A more detailed rubric is here.
On Monday, April 7 you will be given a paper copy of the rubric in the picture above and you will be required to hand it in and assess yourself before I do.
Don't forget to tag your picture: pc40sw08 and trigonometry.
Monday, March 31, 2008
flickr Assignment
Your flickr assignments are due Friday, April 11. Click the picture below to see the rubric that describes how you will be assessed:
Friday, March 28, 2008
New Unit: Exponential Functions
Today was a new day, and today was a new unit. The unit was called exponential functions. Although we didn't do much on the new unit, we were introduced by trying to find 4 different ways to write the following numbers: 2, 3, 4/9, 1/4 with exponents. For an example Mr. Kuropatwa gave us two ways to write the number 2 in exponential form: 2^1 and (1/2)^-1.
While working on these questions we got all are previous tests and quizzes back. Everyone was pretty much down on their marks. Naturally, nice Mr. Kuropatwa discussed marks and how the number on your report card isn't very accurate because that was a number reflecting your knowledge at that point in time, and a given number of days or weeks later it would obviously be different, because in those number of days or weeks you most likely obtain a fuller understanding and a more complete idea on that one concept that you didn't do so well on at first.
Mr. Kuropatwa then talked about "How to be an Expert" http://expertvoices.blogspot.com/search/label/Assignment
It consisted of people who would try a certain thing or try to grasp a concept and realize that they can't do it so they give up. These people were considered to be in the "suck threshold". Then there came the amateur who would figure out how to do something, but just do it the same way over and over, not really ever moving anywhere just staying at that one point. Experts are different then these people because after they figure out this certain concept, they continuously try to improve on it and get better and better. These people are the true experts.
Leading to the end of class, we were given a project that can either be completed solo, in a pair, or with 3 members in a group. It was an assignment on the creation of 4, 5, or 6 math questions depending on the number of people in each group. With a soloist creating 4 questions, a pair creating 5 questions and so on. That required you to explain how you created each problem, and how you solve it step by step. These questions should expand over 2 different units and these should be units that you have had trouble on, in order to expand your knowledge even further. This whole assignment is worth 20% and a due date of your choice, with a final due date of around mid June. Sample problems which reflect the questions to be made for the final projects have to be chosen by April 7, 2008. These can be COPIED from sources such as your excerise book. The picture with trigonometric concepts, such as reflection, waves, etc. is also due by this date. That is all we did in pre-calculus class on the last day of school before spring break. Everyone enjoy their holiday!
Go YELLOOWWW! I'm Out.
-Francis
While working on these questions we got all are previous tests and quizzes back. Everyone was pretty much down on their marks. Naturally, nice Mr. Kuropatwa discussed marks and how the number on your report card isn't very accurate because that was a number reflecting your knowledge at that point in time, and a given number of days or weeks later it would obviously be different, because in those number of days or weeks you most likely obtain a fuller understanding and a more complete idea on that one concept that you didn't do so well on at first.
Mr. Kuropatwa then talked about "How to be an Expert" http://expertvoices.blogspot.com/search/label/Assignment
It consisted of people who would try a certain thing or try to grasp a concept and realize that they can't do it so they give up. These people were considered to be in the "suck threshold". Then there came the amateur who would figure out how to do something, but just do it the same way over and over, not really ever moving anywhere just staying at that one point. Experts are different then these people because after they figure out this certain concept, they continuously try to improve on it and get better and better. These people are the true experts.
Leading to the end of class, we were given a project that can either be completed solo, in a pair, or with 3 members in a group. It was an assignment on the creation of 4, 5, or 6 math questions depending on the number of people in each group. With a soloist creating 4 questions, a pair creating 5 questions and so on. That required you to explain how you created each problem, and how you solve it step by step. These questions should expand over 2 different units and these should be units that you have had trouble on, in order to expand your knowledge even further. This whole assignment is worth 20% and a due date of your choice, with a final due date of around mid June. Sample problems which reflect the questions to be made for the final projects have to be chosen by April 7, 2008. These can be COPIED from sources such as your excerise book. The picture with trigonometric concepts, such as reflection, waves, etc. is also due by this date. That is all we did in pre-calculus class on the last day of school before spring break. Everyone enjoy their holiday!
Go YELLOOWWW! I'm Out.
-Francis
Thursday, March 27, 2008
The Bourne.. ~ er ~ BOB Identity
Rence here reporting his bob a week early before the test takes place but HEY! Let's get this out of the way. I found this to be quite an interesting unit. The whole other identities of trig functions and all that. At first, I was a bit confused but later it all unravelled with the collories, difference of square identities and proving identities.
At times I found proving identities hard because well, like a normal case, you'd have to collect evidence to prove the identitie is real. Very rarely is the identity false because you'd algebraically massage it until it all falls into place like a puzzle, but a sudoku puzzle to be exact, because you're not always going to make a sudoku puzzle perfect from the start. You play around with it here and there. I found playing around with the identities a bit tedious, but it had to be done. It's just that I don't notice certain identities at first that would lead to the final answer.
This was a good unit in my say, theres were just a few frustration points where I had to get over the hump so yeah. I don't and do look forward to the test at the same time. So yeah.
~Rence OUT!
At times I found proving identities hard because well, like a normal case, you'd have to collect evidence to prove the identitie is real. Very rarely is the identity false because you'd algebraically massage it until it all falls into place like a puzzle, but a sudoku puzzle to be exact, because you're not always going to make a sudoku puzzle perfect from the start. You play around with it here and there. I found playing around with the identities a bit tedious, but it had to be done. It's just that I don't notice certain identities at first that would lead to the final answer.
This was a good unit in my say, theres were just a few frustration points where I had to get over the hump so yeah. I don't and do look forward to the test at the same time. So yeah.
~Rence OUT!
JabbaMatheez - The Touch Up.
APPARENTLY I got 1000 marks for finding 'X' SO, I guess I don't have to come to class anymore. =P
We did a little touch up class before we headed on to the next unit. On top of that, we conversed about various things, like when we should have the math test, (Some crazy guy suggested we do the test THEN the pre-test. What a guy huh?), DJ K's baby, Justus' remix to "Apologize" and many other random things... that unfortunately had nothing to do with math. BUT Hupsha, hupsha, quick like a bunny, we went off to work.
If you look at the second slide, you'll see we did a simple warm up and proved some identities. REMEMBER! When the question asks that you PROVE and I mean PROVE, create your GREAT WALL OF JABBA! Which is to say YOU CANNOT CROSS THAT LINE! But when it says SOLVE then you may cross that line and do whatever you wish. That first one on the left side of the slide was done none other than ELVEN. Whom was remarked as... HAWT. So anyways, he simply changed 1 + tan² x into seceant, which is 1/cos² x. He then multiplied the sin²x which would end up becoming tan² x. And so both sides are the same.
On the right side, ben started off by changing seceant and coseceant into their 1 / sine and 1 / cosine bretheren. On the other side of the Great Wall of Jabba, he changed it to 1 / cos² xsin²x, so that when he finished the left side they would be identical. QED!
Intermission...
And Now Back to our Regular Programming...
Closed Captioning brought to you by... Rence
This Blog is sponsored by DJ K!!!
Now if you look at the 3rd slide, it says solve so what does that mean class!? That's right! Mr. K's buying us donuts for tomorrows class! LOL I wish but actually it means you can cross the Great Wall of Jabba. And that's exactly what Thi does in this slide, marked in GREEN. He moves over the 1 and divides 2 from both sides to get cos²x = 1/2. He then square rooted it BUT he forgot the + /- sign. Don't forget that, write that down. Because of that, he missed two of the solutions and so his solution was not completely wrong, but incomplete. Props to Thi for going up to the board and doing that, bcause chances are we would've made the same mistake. Except Ben, but hey, Ben's a genius.
Marked down in BLACK, DJ K's got the elegant way of answering the question. So this is how he breaks it down, and no I don't mean by doing flares and what not. I mean how he takes the 2cos² x - 1 and breaks it down to (√2cosx + 1)(√2cosx - 1) = 0.
From that we can derive that cosx = 1/√2 and cosx = -1/√2 but WWWAAAIIITTT! We have to rationalize that! So it by multiplying both the top and bottom by √2 we get cosx = √2/2, -√2/2, and that's how DJ K get's the other two solutions. Ya'll dig or what?
Continuing on to slide 4, we have a similar question, but sine and cos are in the same equation(2cos²x = 2 + sinx)!
Oh No! Like DJ K said, we couldn't probably do it weeks ago and say "DJ K, I'm sorry but I only work with one way equations. Throw in Sine and Cosine, sorry, no can do." And he's right, but us, being smarter than the average bear find that cos²x = 1 - sin²x. So now everything's sin²x. Can we do it? Yes we can!
So then we basically get everything off to one side in this equation (is now 2(1 - sin²x) = 2 + sinx --> 2 - 2sin²x = 2 + sinx *since the 2's cancel* --> 0 = 2sin²x + sinx) we can then factor out sinx (sinx(2sinx + 1)) in which we find that sinx = - 1/2.
Well, we find through that, that the solution is 11π/6 + 2 kπ ; KЄ I and 7π/6 + 2kπ ; KЄ I. On the next slide we practically do the same thing. Different trig functions and numbers is all.
Another intermission
Continuing to our next slide, we continue with our regular scheduled programming.
In this slide we are given sinα = 4/5 and cos β = -5 / 13 where cosα <> 0. So to find those, we know that sinβ and cosα are in quadrant 2 using Pythagoras's theorem & a picture of the unit square like so.
5² - 4² = 3² (25 - 16 = 9 --> √9 = 3) and 13² - 5² = 12 ² (169 - 25 = 144 --> √144 = 12)
It then asks us to find what Tan(α - β) is so we basically combine Sin(α - β) /Cos(α - β)
GREAT! So now we have cosα = -3/5 and sinβ = 12 / 13. So Now we can get Tan(α - β) .
I know it's homework, but I'll post up my answer so that others can compare. Correct me if I'm wrong PLEASE!
Sin(α - β) = SinαCosβ - CosαSinβ. Cos(α - β) = CosαCosβ + SinαSinβ.
Then, set it up like this. SinαCosβ - CosαSinβ/CosαCosβ + SinαSinβ.
Replace them. (4/5)(-5/13) - (-3/5)(12/13) / (-3/5)(-5/13) + (4/5)(12/13) --> (-20/65) - (36/65) / (15/65) + (48/65) --> (-56/65) / (63/65)
*MULTIPLY BY THE RECIPROCAL* (-56/65) * (65/63)
* 65's REDUCE* -56 / 63. So I got an answer of Tan(α - β) = -56 / 63
That's all for today folks! Remember, we start a new unit tomorrow!!
P.S. HEY fellow JabbaMatheez! We still have our PICTURES due after the break so don't forget that while you're all having fun and having adventures on SPRING BREAK '08!
Oh wait, there's still the masking of the next Scribe!
I will pass the Jabba Scribe on to... FRANCIS!!
~Rence OUT!
Wednesday, March 26, 2008
BOB for Identities
Wow it's already time for another bob?!
Out of all the units, I like this the best for some reason. That doesn't mean I didn't struggle with it though. The first time I saw those proof questions was not pretty. It didn't help that I couldn't remember the formulas or derive them either. I always thought I'd never be able to solve those kinds of questions. But with practice and finally figuring out how to derive, I slowly started getting the answers right. I also found that what works for solving identities is just experimenting at first when you don't see the answer yet. With that said, I just have to work on solving these questions faster because there's only an hour or so to do a test and to find the clever idea in solving them.
Now what I really liked in this unit was the Sine dance. At first I was like "How's this going to help me?" but it did. The sine and cose formula will never leave my mind and I'll always remember it when I need it. That's why I find that I'm really confident when doing the sum and difference questions.
Yeah so the test is on Friday. Just have to get it over with and then it'll be Spring break!
Out of all the units, I like this the best for some reason. That doesn't mean I didn't struggle with it though. The first time I saw those proof questions was not pretty. It didn't help that I couldn't remember the formulas or derive them either. I always thought I'd never be able to solve those kinds of questions. But with practice and finally figuring out how to derive, I slowly started getting the answers right. I also found that what works for solving identities is just experimenting at first when you don't see the answer yet. With that said, I just have to work on solving these questions faster because there's only an hour or so to do a test and to find the clever idea in solving them.
Now what I really liked in this unit was the Sine dance. At first I was like "How's this going to help me?" but it did. The sine and cose formula will never leave my mind and I'll always remember it when I need it. That's why I find that I'm really confident when doing the sum and difference questions.
Yeah so the test is on Friday. Just have to get it over with and then it'll be Spring break!
Trigonometric Identities: Double Identites
The slides:
Link to the slides post.
Link to the slides page.
Hey guys, its Paul here posting my scribe for our class on Double Identities on Tuesday (just in time amirite?).
So we started off the class with a review of the previous class on slide 2. The basic concept here is that the distance between Q and P is equal to cos(a-b).
In slide 3, we used this idea to find the Sum Identity with Even and Odd functions since we already have the Difference Identity.
Sum Identity: cos(a+b) = cosacosb - sinasinb
Difference Identity: cos(a-b) = cosacosb + sinasinb
Cosine is an even function, sine and tangent are odd.
We use the same concept for finding the Sum and Difference identities of the distance between R and Q on slides 4 and 5.
Then we take a break from our "regularly scheduled programming" (I wonder if that counts as one of Mr.K's catchphrases that we should put on the board) with a short and simple quiz on slides 7, 8 and 9.
Slide 7: Simplify the expressions. Pretty self explanatory, use identities to find the simplest expression.
Slide 8: Prove the identities. Again, stuff we've done before.
Slide 9: Find the exact value of sin(5pi/12). This one took a little more work, but it was actually easy because the values a = pi/6 and b = pi/4 were given to us. As Mr.K explained, on tests we will simply be given a value to find with a formula, which means we'll have to find a and b on our own. Since they're given to us, we can just plug the values into the formula and get our answer.
As we return to our regularly scheduled programming on slide 10, we are given and identity to prove. The Sum Identity for tangent, to be precise.
The solution is pretty long and looks complicated but it basically follows these steps:
tan(a+b) = sin(a+b)/cos(a+b)
And since we already know what sin(a+b) and cos(a+b) equal to, we get a really long equation (one I won't bother to type since its right there in the slides). Once you have your really long equation, most of it simplifies or reduces to the proper solution of tana+tanb/1-tanatanb.
And since Math is the science of patterns, you'll probably know what we're going to do next. Next, we find the Difference Identity of tan (tan(a-b)) since we have the Sum Identity. We do this on Slide 11 in an equally long but rewarding process.
In summary:
tan(a+b) = tana+tanb/1-tanatanb
tan(a-b) = tana-tanb/1+tanatanb
On Slides 12 and 13 we learn about Double Angle Identities (which is different from a double identity). Here we apply some old stuff to a new problem, and turn something like:
sin(2theta)
Into:
sin(theta + theta)
Which would look more familiar as say... sin(a+b)?
Since its exact same thing (so long as b = a), we can rewrite sin(2theta) as:
(sintheta)(costheta) + (costheta)(sintheta)
And then simplify it so it looks like
2sintheta(costheta)
We then apply this concept to find the Double Angle Identities of cos and tan.
sin(2theta) = 2sintheta(costheta)
cos(2theta) = cos^2(theta) - sin^2(theta), 1-2sin^2(theta), 2cos^2(theta) -1
tan(2theta) = 2tantheta/1-tan^2(theta)
And that basically sums up what we did that day in class.
Now I'd like to add a small note to the sine dance to make provisions for the tan identities, which is obviously a pain to figure out via sin(a+b) and cos(a+b). Therefore, I propose a tan dance to help us remember the tan identities.
Since this is all in terms of tangent, we do the dance in variable order, something like this:
Alpha, Beta, Divide, 1, Bust-a-move, Alphabeta
or in stick man form...
Im just putting it out there. Seems easier than solving sin(a+b)/cos(a+b) every time you want to do a tan identity to me.
And that concludes my painfully brief scribe post about Double Identities. Hope you guys get a good nights sleep and enjoy our last day before SPRING BREAK.
Yeah man.
Link to the slides post.
Link to the slides page.
Hey guys, its Paul here posting my scribe for our class on Double Identities on Tuesday (just in time amirite?).
So we started off the class with a review of the previous class on slide 2. The basic concept here is that the distance between Q and P is equal to cos(a-b).
In slide 3, we used this idea to find the Sum Identity with Even and Odd functions since we already have the Difference Identity.
Sum Identity: cos(a+b) = cosacosb - sinasinb
Difference Identity: cos(a-b) = cosacosb + sinasinb
Cosine is an even function, sine and tangent are odd.
We use the same concept for finding the Sum and Difference identities of the distance between R and Q on slides 4 and 5.
Then we take a break from our "regularly scheduled programming" (I wonder if that counts as one of Mr.K's catchphrases that we should put on the board) with a short and simple quiz on slides 7, 8 and 9.
Slide 7: Simplify the expressions. Pretty self explanatory, use identities to find the simplest expression.
Slide 8: Prove the identities. Again, stuff we've done before.
Slide 9: Find the exact value of sin(5pi/12). This one took a little more work, but it was actually easy because the values a = pi/6 and b = pi/4 were given to us. As Mr.K explained, on tests we will simply be given a value to find with a formula, which means we'll have to find a and b on our own. Since they're given to us, we can just plug the values into the formula and get our answer.
As we return to our regularly scheduled programming on slide 10, we are given and identity to prove. The Sum Identity for tangent, to be precise.
The solution is pretty long and looks complicated but it basically follows these steps:
tan(a+b) = sin(a+b)/cos(a+b)
And since we already know what sin(a+b) and cos(a+b) equal to, we get a really long equation (one I won't bother to type since its right there in the slides). Once you have your really long equation, most of it simplifies or reduces to the proper solution of tana+tanb/1-tanatanb.
And since Math is the science of patterns, you'll probably know what we're going to do next. Next, we find the Difference Identity of tan (tan(a-b)) since we have the Sum Identity. We do this on Slide 11 in an equally long but rewarding process.
In summary:
tan(a+b) = tana+tanb/1-tanatanb
tan(a-b) = tana-tanb/1+tanatanb
On Slides 12 and 13 we learn about Double Angle Identities (which is different from a double identity). Here we apply some old stuff to a new problem, and turn something like:
sin(2theta)
Into:
sin(theta + theta)
Which would look more familiar as say... sin(a+b)?
Since its exact same thing (so long as b = a), we can rewrite sin(2theta) as:
(sintheta)(costheta) + (costheta)(sintheta)
And then simplify it so it looks like
2sintheta(costheta)
We then apply this concept to find the Double Angle Identities of cos and tan.
sin(2theta) = 2sintheta(costheta)
cos(2theta) = cos^2(theta) - sin^2(theta), 1-2sin^2(theta), 2cos^2(theta) -1
tan(2theta) = 2tantheta/1-tan^2(theta)
And that basically sums up what we did that day in class.
Now I'd like to add a small note to the sine dance to make provisions for the tan identities, which is obviously a pain to figure out via sin(a+b) and cos(a+b). Therefore, I propose a tan dance to help us remember the tan identities.
Since this is all in terms of tangent, we do the dance in variable order, something like this:
Alpha, Beta, Divide, 1, Bust-a-move, Alphabeta
or in stick man form...
Im just putting it out there. Seems easier than solving sin(a+b)/cos(a+b) every time you want to do a tan identity to me.
And that concludes my painfully brief scribe post about Double Identities. Hope you guys get a good nights sleep and enjoy our last day before SPRING BREAK.
Yeah man.
Trignometry Identities for March 26.
First off! Today was formal day! Hopefully you dressed formally, if not, shame! Just kidding, math doesn't require formal clothes. I'm not sure if I should have waited for Paul to scribe for yesterdays class, but today he chose me for scribe while we were in class.
Today wasn't very exciting, and you all know I'm not talking about the talent show, now that was INSANE. Yeah it was pretty sweet, but you know what else is sweet? CANDY! Okay, but anyways. Today in class we had a substitute teacher. The only thing we did in class was a worksheet entitled "Trig Identities Worksheet 3.4" and as you might have guessed it's about solving trigonometry identities. This worksheet was really quite helpful in understanding some identities. It Cleared some of my thoughts at least. In total there were 16 questions on this worksheet, and some questions were simple and others gave you the "where do I start?!" question. After all, this worksheet did take the whole class and was plenty of time to finish some of those tough ones. If you're not done the worksheet, it was for homework, and I'm sure that most of us had to bring it home. It was really helpful and good luck on the pre-test tomarrow.
Like Jeff out of the Coffin! I'm Out!
-Francis
Today wasn't very exciting, and you all know I'm not talking about the talent show, now that was INSANE. Yeah it was pretty sweet, but you know what else is sweet? CANDY! Okay, but anyways. Today in class we had a substitute teacher. The only thing we did in class was a worksheet entitled "Trig Identities Worksheet 3.4" and as you might have guessed it's about solving trigonometry identities. This worksheet was really quite helpful in understanding some identities. It Cleared some of my thoughts at least. In total there were 16 questions on this worksheet, and some questions were simple and others gave you the "where do I start?!" question. After all, this worksheet did take the whole class and was plenty of time to finish some of those tough ones. If you're not done the worksheet, it was for homework, and I'm sure that most of us had to bring it home. It was really helpful and good luck on the pre-test tomarrow.
Like Jeff out of the Coffin! I'm Out!
-Francis
IDENTITIES: "Hi, my name is Bob"
So... here I am again, BOBBING for apples. Identities was really a short unit and breezed by so fast... or it appears most other people did. I had to and still am fighting through the anything-but-mild wind.
As usual, I'm having problems. Not with finding the identities per se, but I have a problem with FINDING the identities. I'm [math] exercising like crazy and trying to keep up with everyone...and not to mention Mr. K. He could practically save lives if they depended on him doing identities. But me...I take forever!! FOREVER! And it's frustrating because when I do my BOB's I'm always panicking because I feel behind. And I'm not sure if it's my work ethic...but I'm just not absorbing information fast enough...and it's the same thing before any test I've had this semester. I have to find a way to fix it. Because I know I failed miserably on the last test... my own fault. I accept that. But this post is getting rather depressing now and very unproductive, and I admit, a bit unprofessional. But I think that's the only issue I have with this unit. I don't always figure out which side of the identity to "massage" and I just don't see it. It's nothing a little practice couldn't fix.
As usual, I'm having problems. Not with finding the identities per se, but I have a problem with FINDING the identities. I'm [math] exercising like crazy and trying to keep up with everyone...and not to mention Mr. K. He could practically save lives if they depended on him doing identities. But me...I take forever!! FOREVER! And it's frustrating because when I do my BOB's I'm always panicking because I feel behind. And I'm not sure if it's my work ethic...but I'm just not absorbing information fast enough...and it's the same thing before any test I've had this semester. I have to find a way to fix it. Because I know I failed miserably on the last test... my own fault. I accept that. But this post is getting rather depressing now and very unproductive, and I admit, a bit unprofessional. But I think that's the only issue I have with this unit. I don't always figure out which side of the identity to "massage" and I just don't see it. It's nothing a little practice couldn't fix.
BOB for identities
Identities! First I would like to say that I missed 2 classes in this unit. Why? NO CLUE. But it's not good! Every class is needed for me, and makes things more difficult and tuning into it is hard. Thank you.
What I think about this unit is that it's simple and easy as long as you UNDERSTAND IT. Which is what Mr. K has done. Explained each formula. It takes some thinking, but the steps are similar is how I see it. I'm still unsure of some of the things, like when you know what Cosa and SinB I forget how to find CosB and Sina. But I'm actually BOB'ing before I'm doing my exercises so It may be to early to dicuss what I'm having difficulty on. It wouldn't hurt to have one more workshop. This unit was very fun.
What I think about this unit is that it's simple and easy as long as you UNDERSTAND IT. Which is what Mr. K has done. Explained each formula. It takes some thinking, but the steps are similar is how I see it. I'm still unsure of some of the things, like when you know what Cosa and SinB I forget how to find CosB and Sina. But I'm actually BOB'ing before I'm doing my exercises so It may be to early to dicuss what I'm having difficulty on. It wouldn't hurt to have one more workshop. This unit was very fun.
Tuesday, March 25, 2008
Trigonometric Identities Practice
I can see that most of you guys are having some trouble with trigonometric identities. The only way that you can be good at it is by doing some practice questions, so you can develop a knack on seeing the patterns in this mini mind puzzles. There are many good sites out there that you can practice on, so I decided to make a list of few sites that will definitely help all of you. Here it goes:
http://library.thinkquest.org/20991/alg2/trigi.html#Quiz
http://www.syvum.com/cgi/online/serve.cgi/math/trigo/trig2.tdf?0
http://www.quia.com/cc/448321.html
http://www.epcc.edu/Student/Tutorial/Mathcenter/Exams/trigquizzes/quiz14/QUIZ14.htm
http://www.libraryofmath.com/trigonometric-identities-quiz.html
http://www.batesville.k12.in.us/physics/CalcNet/Trig_Review/Trig_Review.html
http://college.hmco.com/cgi-bin/SaCGI.cgi/ace1app.cgi?FNC=AcePresent__Apresent_html___mathematics_larson_algebra_trig_6e_07-02
http://library.thinkquest.org/20991/alg2/trigi.html#Quiz
http://www.syvum.com/cgi/online/serve.cgi/math/trigo/trig2.tdf?0
http://www.quia.com/cc/448321.html
http://www.epcc.edu/Student/Tutorial/Mathcenter/Exams/trigquizzes/quiz14/QUIZ14.htm
http://www.libraryofmath.com/trigonometric-identities-quiz.html
http://www.batesville.k12.in.us/physics/CalcNet/Trig_Review/Trig_Review.html
http://college.hmco.com/cgi-bin/SaCGI.cgi/ace1app.cgi?FNC=AcePresent__Apresent_html___mathematics_larson_algebra_trig_6e_07-02
BOB Version 3: Trigonometric Identities
I found this unit less daunting than the other previous units we've done so far. (This is because we don't have to memorize something huge that's like worth 1 megabyte of our memory, such as the unit circle, or get anxious over a graphing portion of a unit--because there is no graphing portion in this unit.) Comparing topics in this unit, I liked the "prove the following identity" part because it seems to be the easiest topic to absorb. Remembering how to do the sine dance (which aided us in remembering how to remember the 'dancing equations'--that's what I call them) helped out A LOT, and so, I found doing those questions without difficulty.
As of now, I have to go over the slides for the proofs of sum and differences because I haven't yet looked at them! I'm not too comfortable with questions where the questions tell what alpha and beta are, then we would have to draw a circle to find out quadrants they're in, then use the dancing equations to determine where a point is in which quadrant--because right now I'm having trouble wrapping my head around the 'setting up' part where we have to draw the circle to see what angle alpha, beta, a, or b are in. If you don't know what I'm talking about, then refer to March 20's slide #4 and 5, please and thank you! But now, I've just recently had a moment of clarity and doing those kinds of questions make sense to me.
Also, one of the questions in our homework that was assigned, EXERCISE 16: SUM AND DIFFERENCE IDENTITIES I, #9, it tells us to graph sin (t + 3pi/2) = -cos t, in which I have no idea how to graph that. And there maybe more questions that I'm not comfortable in doing because I haven't looked at them yet.
The test on trig. identities is set for Friday, and I'm sure I can prepare myself before that day. So 'til then, may we all grok in fullness!
P.S. Is it just me or did Mr.K not posted up today's slides? I was planning on using those slides to study off of because I was going to look over those slides thinking it would be more convenient for me to study off of since I am on the computer now typing up my blog post.
As of now, I have to go over the slides for the proofs of sum and differences because I haven't yet looked at them! I'm not too comfortable with questions where the questions tell what alpha and beta are, then we would have to draw a circle to find out quadrants they're in, then use the dancing equations to determine where a point is in which quadrant--because right now I'm having trouble wrapping my head around the 'setting up' part where we have to draw the circle to see what angle alpha, beta, a, or b are in. If you don't know what I'm talking about, then refer to March 20's slide #4 and 5, please and thank you! But now, I've just recently had a moment of clarity and doing those kinds of questions make sense to me.
Also, one of the questions in our homework that was assigned, EXERCISE 16: SUM AND DIFFERENCE IDENTITIES I, #9, it tells us to graph sin (t + 3pi/2) = -cos t, in which I have no idea how to graph that. And there maybe more questions that I'm not comfortable in doing because I haven't looked at them yet.
The test on trig. identities is set for Friday, and I'm sure I can prepare myself before that day. So 'til then, may we all grok in fullness!
P.S. Is it just me or did Mr.K not posted up today's slides? I was planning on using those slides to study off of because I was going to look over those slides thinking it would be more convenient for me to study off of since I am on the computer now typing up my blog post.
Identities and BOB
The unit was fairly short, and at first it seemed quite easy. Later on is a different story unfortunately. I was pretty much confused when all these other equations came into the picture. The sine dance was pretty cool, it helped to clear up some of my confusion, but nonetheless identities was quite tough. It's pretty much hit and miss for me. If i have a good day, the anwser will just pop out of my head, first time around. Other days are just random though. I have to try one side, than the other, then try multiple times for each side. The answer just doesn't come to me. As I watch Mr. Kuropatwa write the answer on the board, in like..30 seconds. My brain does a double backflip and I realize how easy it actually was. That is why this unit is pretty much hit and miss for me. Double angle identities are still killing me though. the tangent equations are also slightly blurry, but I just hope it all comes into my head for the test! Good luck everyone!
Leaning with Wiggles! I'm out!
-Francis
Leaning with Wiggles! I'm out!
-Francis
BOB: Identities
So we come to the end of the Identity unit, therefore we must all bob before the test which I believe will be on Friday..!
Overall I believe this unit was easy compared to the previous unit. However, I still think it's not THAT easy because of all of the algebra we have to do. Like any other unit, there are parts where I think are easier than others. In the beginning of the unit, I thought that this unit would be fairly easy because the questions were pretty much straight forward. As we moved forward into the unit it became a bit confusing starting with the sum and difference identities. I still have a bit of trouble because I always have to take a minute to recall the different equations to use. In today's class we learned the final part of the unit, double angle identities which I found pretty straight foward.
I think my biggest problems I have to fix will be not trying to skip any steps because once I start doing that, I end up messing up the whole problem. I also need to look over my work to check if my algebra is correct as well as finding patterns quicker because I tend to waste time trying to figure out certain things.
With that said, I hope everyone does well on the test and don't forget the sine dance! LoL Goodluck everyone! Bye!
Overall I believe this unit was easy compared to the previous unit. However, I still think it's not THAT easy because of all of the algebra we have to do. Like any other unit, there are parts where I think are easier than others. In the beginning of the unit, I thought that this unit would be fairly easy because the questions were pretty much straight forward. As we moved forward into the unit it became a bit confusing starting with the sum and difference identities. I still have a bit of trouble because I always have to take a minute to recall the different equations to use. In today's class we learned the final part of the unit, double angle identities which I found pretty straight foward.
I think my biggest problems I have to fix will be not trying to skip any steps because once I start doing that, I end up messing up the whole problem. I also need to look over my work to check if my algebra is correct as well as finding patterns quicker because I tend to waste time trying to figure out certain things.
With that said, I hope everyone does well on the test and don't forget the sine dance! LoL Goodluck everyone! Bye!
Finding the Identity of BOB!!!
Hi this is benofschool and this is my BOB. This unit seems to be the most difficult thus far. I can grasp most of the content but I just need to perfect my algebra to fully possess the skills needed to pass the Identities Unit. I believe I need just a tad more practice so I will finish all of the exercises that were relevant to the unit and also work on any questions from class that I previously haven't solved in my notes. I guess that should be enough. My biggest problem though would be my habit of overlooking things. I might miss a clever idea or step in proving identities. I will have to watch out for that. That is all for this unit's BOB and until next time good luck on the pretest and test and have a great spring break if I am not chosen as a scribe before the break!!! For practice I recommend this site: www.math40s.com. :)
BOB the Blogger is Back! (Identities Unit)
Its that time of the unit again, where we must do our BOBs!. thus I commence with mine own BOB for Identities.
So to begin I'd like to say that this unit was easily the most up and down one for me. I'll explain what that means exactly right now. With most of the other units, it was either steadily easy, or steadily difficult. However, with the identities unit, I found that it was all over the place. Some of it I got extremely quickly, and other parts of it I struggled profoundly with.
The easiest parts I think, involved the use of the sum and difference identities. Substituting the values of sine or cosine, or substituting the identity itself seemed relatively straightforward to me.
One of the more difficult parts, involved solving identities in general, as I'd usually get stuck in the paralyzed frozen mode Mr. K was talking about because I couldnt see the end of the problem, and thus, didnt want to advance into an uncertainty. I am working on forcing myself to continue and try new things though to get rid of that bad habit :P
One of the things I found most helpfull was the sine dance. Expect to see me doing it during the test :D
Well I think thats all, so Ciao!
So to begin I'd like to say that this unit was easily the most up and down one for me. I'll explain what that means exactly right now. With most of the other units, it was either steadily easy, or steadily difficult. However, with the identities unit, I found that it was all over the place. Some of it I got extremely quickly, and other parts of it I struggled profoundly with.
The easiest parts I think, involved the use of the sum and difference identities. Substituting the values of sine or cosine, or substituting the identity itself seemed relatively straightforward to me.
One of the more difficult parts, involved solving identities in general, as I'd usually get stuck in the paralyzed frozen mode Mr. K was talking about because I couldnt see the end of the problem, and thus, didnt want to advance into an uncertainty. I am working on forcing myself to continue and try new things though to get rid of that bad habit :P
One of the things I found most helpfull was the sine dance. Expect to see me doing it during the test :D
Well I think thats all, so Ciao!
Bob for Identities
Wow i don't know how to begin this but here it goes
i found this unit the harder then the other units for some odd reason solving the identities got me really confused. but the things that helped me out off that confusion are those great workshop classes. For some mysterious odd reason when we have those workshop classes my brain starts to function a lot better. The only thing that gets confusing is that there are too many ways on how to solve a problem other whys its all g. the thing that i absolutely loved about this unit was the sin dance is a great mnemonic on how too remember the equation. this was a hard unit but also a fun unit. i hope that a pre test and test are easy.
i found this unit the harder then the other units for some odd reason solving the identities got me really confused. but the things that helped me out off that confusion are those great workshop classes. For some mysterious odd reason when we have those workshop classes my brain starts to function a lot better. The only thing that gets confusing is that there are too many ways on how to solve a problem other whys its all g. the thing that i absolutely loved about this unit was the sin dance is a great mnemonic on how too remember the equation. this was a hard unit but also a fun unit. i hope that a pre test and test are easy.
BOB: Identities
I found this unit, although short, to be very hard and confusing. It was definitely the hardest unit so far for me and I still have problems with proving identities. I'm okay with everything else but this since they're like logic puzzles, where you have to look closer and try to see what would work. I was very frustrated when we had to prove an identity and then I'd be so lost as to where to start first. I'd try everything that I could think of, only to find out that my answer was way off, and that I had started off in the right way. It was even more of a slap in the face when Mr. K or someone else would post the solution to the identity and then I would be like, "Oh wow! It was that easy?" I admit, I am not as dumbfounded towards solving identities as the first few days, but its still a bit hard and to add to the fact that I'm incredibly slow at doing them. >_<
Anyways, yeah, that's all I've got to say, or rant about. Hopefully I, and of course everyone else, does well on the test on ..Thursday? Well then, that's all for today! *quietly walks out of blog*
Anyways, yeah, that's all I've got to say, or rant about. Hopefully I, and of course everyone else, does well on the test on ..Thursday? Well then, that's all for today! *quietly walks out of blog*
Monday, March 24, 2008
Proofs; The Sum and Difference Identities.
Alright, lets get this show on the road as I'm now the replacement scribe for today. First things first.
PAUL IS THE SCRIBE FOR TOMORROW (March 25th, 2008)
Anyways now that thats dealt with lets begin.
We started class off with a quiz, which for many of us (or at least I think it was many of us) had a bit of trouble getting into. One of the main points to remember in this little intro portion from the quiz was to look at difference of squares, and look out for one of the factors of them. Many of the questions were solved by algebraically massaging the expression using a difference of squares (where applicable.) Anyways the answers to the quiz(and in most cases a step by step guide of how to get to those answers) can be found;
Here
Things to Remember
-Look for a difference of squares, or a factor of a difference of squares
-Remember that you dont need to memorize the corollaries for the trig identities, but rather, derive them yourself when you need them from the Pythagorean theorem. Sine^2 + Cosine ^ 2 = 1.
*Note* In case the images are to small, they're all on the slides, so simply jet on over to the link above and hunt em down :)
So with that little opening bit of the lesson out of the way, lets continue with the main content of this scribe post, The proofs of the Sum and Difference Identities.
If you remember from the previous class on Thursday, we were right on the precipice of uncovering the, "clever idea" behind these identities. If you recall, we worked with the angles ROP, and QOP. Now assuming the angle was in a unit circle, we were able to find the coordinates of points Q, P, and R in terms of Sine and Cosine. After that we implemented the first portion of our clever idea, a rigid transformation of the overall angle, which put it on the x and y axis perfectly. The finding of the angles all lie in the fact that angle QOP, is equivalent to angle alpha subtract angle Beta. Thus when the whole thing is rotated, the angle QOP on the x axis may be described as Alpha - Beta.
Now that our recap is completed lets get into the new stuff! By looking at the above you may see an orange line. This represents the distance between Q and P. This here, is the bread and butter of the clever idea for today. So with the rigid transformation from the first to second diagram, the distance from Q to P must have remained the same (hence rigid transformation.) It is from this point the real magic can be worked. Next we used the Distance formula √[(X2-X1)^2 + (Y2-Y1)^2] (which is the Pythagorean theorem by the way.) and plugged in the values we found before. Thus we got everything seen on the following slide.
Once the Sine and Cosine values have been plugged in you begin to simplify it all down until you have the cosine difference identity. Nice eh? Now because that only covered the difference identity, we have to find the sum identity. To advance here however, we must go back, to Odd and Even functions. If your memory serves you correctly you should remember that Cosine is an Even function and Tangent, and Sine are both Even functions, that is to say that Cosine (x) = Cosine (-x), Tan(-x) = -Tangent(x), and Sine(-x) = -Sine(x). So keeping all that in mind, and remembering we have already proven one identity, we can now prove the cosine sum identity As follows
Cos(α+β) = Cos (α-[-β]) <---------- Clever Idea!
|Cos (α) Cos (-β) + Sin (α) Sin (-β)
|Cos (α) Cos (β) + Sin (α) Sin (β) *
Cos(α+β) = | Cos (α) Cos(β) + Sin (α) Sin (β)
By substituting in the even or odd identity for Sine and Cosine we were able to come up with the proof for the Sum identities.
Now I think that wraps up just about everything we did in this class. Today.
Remember that homework for today is to (using the above) come up with the sum and difference proofs for Sine. To do this use what we learned today, on R'Q' = RQ rather then Q'P' = QP. Also we have Exercise the Next, which is known more specifically as Exercise 17.
So ja, thats it. Hopefully it made sense. In case you missed it up top, paul is scribe for tomorrow. Now I must go build a catapult for physics! -_-;
PAUL IS THE SCRIBE FOR TOMORROW (March 25th, 2008)
Anyways now that thats dealt with lets begin.
We started class off with a quiz, which for many of us (or at least I think it was many of us) had a bit of trouble getting into. One of the main points to remember in this little intro portion from the quiz was to look at difference of squares, and look out for one of the factors of them. Many of the questions were solved by algebraically massaging the expression using a difference of squares (where applicable.) Anyways the answers to the quiz(and in most cases a step by step guide of how to get to those answers) can be found;
Here
Things to Remember
-Look for a difference of squares, or a factor of a difference of squares
-Remember that you dont need to memorize the corollaries for the trig identities, but rather, derive them yourself when you need them from the Pythagorean theorem. Sine^2 + Cosine ^ 2 = 1.
*Note* In case the images are to small, they're all on the slides, so simply jet on over to the link above and hunt em down :)
So with that little opening bit of the lesson out of the way, lets continue with the main content of this scribe post, The proofs of the Sum and Difference Identities.
If you remember from the previous class on Thursday, we were right on the precipice of uncovering the, "clever idea" behind these identities. If you recall, we worked with the angles ROP, and QOP. Now assuming the angle was in a unit circle, we were able to find the coordinates of points Q, P, and R in terms of Sine and Cosine. After that we implemented the first portion of our clever idea, a rigid transformation of the overall angle, which put it on the x and y axis perfectly. The finding of the angles all lie in the fact that angle QOP, is equivalent to angle alpha subtract angle Beta. Thus when the whole thing is rotated, the angle QOP on the x axis may be described as Alpha - Beta.
Now that our recap is completed lets get into the new stuff! By looking at the above you may see an orange line. This represents the distance between Q and P. This here, is the bread and butter of the clever idea for today. So with the rigid transformation from the first to second diagram, the distance from Q to P must have remained the same (hence rigid transformation.) It is from this point the real magic can be worked. Next we used the Distance formula √[(X2-X1)^2 + (Y2-Y1)^2] (which is the Pythagorean theorem by the way.) and plugged in the values we found before. Thus we got everything seen on the following slide.
Once the Sine and Cosine values have been plugged in you begin to simplify it all down until you have the cosine difference identity. Nice eh? Now because that only covered the difference identity, we have to find the sum identity. To advance here however, we must go back, to Odd and Even functions. If your memory serves you correctly you should remember that Cosine is an Even function and Tangent, and Sine are both Even functions, that is to say that Cosine (x) = Cosine (-x), Tan(-x) = -Tangent(x), and Sine(-x) = -Sine(x). So keeping all that in mind, and remembering we have already proven one identity, we can now prove the cosine sum identity As follows
Cos(α+β) = Cos (α-[-β]) <---------- Clever Idea!
|Cos (α) Cos (-β) + Sin (α) Sin (-β)
|Cos (α) Cos (β) + Sin (α) Sin (β) *
Cos(α+β) = | Cos (α) Cos(β) + Sin (α) Sin (β)
By substituting in the even or odd identity for Sine and Cosine we were able to come up with the proof for the Sum identities.
Now I think that wraps up just about everything we did in this class. Today.
Remember that homework for today is to (using the above) come up with the sum and difference proofs for Sine. To do this use what we learned today, on R'Q' = RQ rather then Q'P' = QP. Also we have Exercise the Next, which is known more specifically as Exercise 17.
So ja, thats it. Hopefully it made sense. In case you missed it up top, paul is scribe for tomorrow. Now I must go build a catapult for physics! -_-;
Saturday, March 22, 2008
Proving Identities with cos and sin
Hey everybody, this is Paul, posting the scribe post for our Thursday class which was on the 20th. Sorry this is so late, its been a hectic week.
Note: This is the symbol for theta in my blog post: Θ.
Okay, so Thursday we continued the topic of using identities with sin and cos and expanded it so we could prove stuff like sin(pi/6+Θ) + sin(pi/6-Θ) = cosΘ. This is demonstrated in our 2nd slide.
At first the solution confused me, but I realized the solution works by using the formula we used previously to convert a formula that looks like:
sin(α+β) + sin(α-β) = cosβ
And use formulas we already know to convert it to:
(sinαcosβ + cosαsinβ) + (sinαcosβ - cosαsinβ)
And then reduce to get cosΘ.
In slide 3, we were given a question that asked us to find the cosine of alpha (cosα) and the sin of beta (sinβ). Initially, we got the wrong final answer because we made the mistake of saying that α = -3/5, which is WRONG.
The truth is we never get the value of alpha, we only get the value of the cosine of alpha.
Once we realized our error, we solved the problem properly by using the formula cos(α+β) = cosαcosβ - sinα+sinβ, giving us the correct answer of -33/45. The proper solution is on slide 4.
After that, we were given the same question except we were told to solve for sin(α+β). This takes place on slide 5. The solution is pretty straightforward since you just change the formula but keep all the values you already found in the previous question.
Mr.K then started talking about his "clever idea." He explained to us that the R, Q, and P can be defined in terms of sin and cos on slide 6. He then started to talk about the values of cos and sin in the rotated triangle, but didn't finish because the class ended. I guess we'll find out how clever his idea is on Monday.
Slide 7 shows us how to rotate a point 90 degrees in terms of coordinates. If your coordinates are (x,y) unrotated, then when you rotate it 90 degrees your coordinates will be (-y,x) (such that x = -y and y=x). For 180 degrees, your outcome would be (-x,-y).
And the sums up my scribe post. Sorry it took so long to be posted, especially since its rather short. I didn't manage to get around to doing it until now.
And since nobody's told me they want to be scribe, I will randomly choose a name from a hat (in my mind).
The lucky winner is... ZEPH.
And colour me surprised, the name's not in pink. (gasp)
Somehow it still stands out from the rest of the post.
Good night and farewell.
P.S. And Mr.K, I think the scribe list needs some updating?
Note: This is the symbol for theta in my blog post: Θ.
Okay, so Thursday we continued the topic of using identities with sin and cos and expanded it so we could prove stuff like sin(pi/6+Θ) + sin(pi/6-Θ) = cosΘ. This is demonstrated in our 2nd slide.
At first the solution confused me, but I realized the solution works by using the formula we used previously to convert a formula that looks like:
sin(α+β) + sin(α-β) = cosβ
And use formulas we already know to convert it to:
(sinαcosβ + cosαsinβ) + (sinαcosβ - cosαsinβ)
And then reduce to get cosΘ.
In slide 3, we were given a question that asked us to find the cosine of alpha (cosα) and the sin of beta (sinβ). Initially, we got the wrong final answer because we made the mistake of saying that α = -3/5, which is WRONG.
The truth is we never get the value of alpha, we only get the value of the cosine of alpha.
Once we realized our error, we solved the problem properly by using the formula cos(α+β) = cosαcosβ - sinα+sinβ, giving us the correct answer of -33/45. The proper solution is on slide 4.
After that, we were given the same question except we were told to solve for sin(α+β). This takes place on slide 5. The solution is pretty straightforward since you just change the formula but keep all the values you already found in the previous question.
Mr.K then started talking about his "clever idea." He explained to us that the R, Q, and P can be defined in terms of sin and cos on slide 6. He then started to talk about the values of cos and sin in the rotated triangle, but didn't finish because the class ended. I guess we'll find out how clever his idea is on Monday.
Slide 7 shows us how to rotate a point 90 degrees in terms of coordinates. If your coordinates are (x,y) unrotated, then when you rotate it 90 degrees your coordinates will be (-y,x) (such that x = -y and y=x). For 180 degrees, your outcome would be (-x,-y).
And the sums up my scribe post. Sorry it took so long to be posted, especially since its rather short. I didn't manage to get around to doing it until now.
And since nobody's told me they want to be scribe, I will randomly choose a name from a hat (in my mind).
The lucky winner is... ZEPH.
And colour me surprised, the name's not in pink. (gasp)
Somehow it still stands out from the rest of the post.
Good night and farewell.
P.S. And Mr.K, I think the scribe list needs some updating?
Thursday, March 20, 2008
Wednesday, March 19, 2008
Sum and Difference Identities
Hey Class, it's me again scribing for you. In this particular class, we took a kinesthetic approach in learning about identities. We were given 4 formulas, although Mr. K prefers not to use formulas and would rather have us understand where those formulas are derived from instead. The formulas seemed like quite a task to memorize so... we had a little help. From the greatest mathematical dance of all time:
THE SINE DANCE.
Sin(α + β) = SinαCosβ + CosαSinβ
Sin(α - β) = SinαCosβ - CosαSinβ
Cos(α + β) = CosαCosβ - SinαSinβ
Cos(α - β) = CosαCosβ + SinαSinβ
This dance contains three easy steps which are called: SINE, COSINE AND BUST A MOVE! The order of which the steps are performed are located in the formulas shown above. In the first step sine, which is used to represent all the sine functions, we stick our arms out with one behind our back and bend them so that it forms the letter "s". The next step cosine, is even more simple than the first and is used to represent all the cosine. All you have to do is make the letter "c" with your arms. The final step is used to represent the sign change that cosine makes. To perform this step, you rotate your body 180 degrees and then hold your arms in front of your chest and make the letter "x". Once you have those down, feel free to use it as often as you like to aid you prove the sum and difference identities because that's what we did next except without the dancing.
Well, my scribe post is over now so that means I get to choose the next scribe. The next scribe is Paul. This begins cycle three.
By the way, I found about that writing your name in red ink thing. It turns that that superstition is Japanese. Still not sure if bad luck comes to the writer or the person who's name is written.
THE SINE DANCE.
Sin(α + β) = SinαCosβ + CosαSinβ
Sin(α - β) = SinαCosβ - CosαSinβ
Cos(α + β) = CosαCosβ - SinαSinβ
Cos(α - β) = CosαCosβ + SinαSinβ
This dance contains three easy steps which are called: SINE, COSINE AND BUST A MOVE! The order of which the steps are performed are located in the formulas shown above. In the first step sine, which is used to represent all the sine functions, we stick our arms out with one behind our back and bend them so that it forms the letter "s". The next step cosine, is even more simple than the first and is used to represent all the cosine. All you have to do is make the letter "c" with your arms. The final step is used to represent the sign change that cosine makes. To perform this step, you rotate your body 180 degrees and then hold your arms in front of your chest and make the letter "x". Once you have those down, feel free to use it as often as you like to aid you prove the sum and difference identities because that's what we did next except without the dancing.
Well, my scribe post is over now so that means I get to choose the next scribe. The next scribe is Paul. This begins cycle three.
By the way, I found about that writing your name in red ink thing. It turns that that superstition is Japanese. Still not sure if bad luck comes to the writer or the person who's name is written.
Tuesday, March 18, 2008
More Pythagorean Identities
K hi guys. My turn to blog for the day or should i say...night.
So we started the morning class off by watching that article 13 video. After that it was straight to math. Today's math class was a workshop since many of us enjoy doing this activity.
Anyways, the lesson for today had a great deal of emphasis on...elegance in showing our work. By elegance, I mean the quickest and most direct way of solving or proofing an identity. The formulas above help a great deal in achieving this elegance. To make things elegant we can change everything into sine, cosine, or any of the other identies to solve faster. Like the question on slide 3. Work did not need to be shown because it is already proven that tan^2a is = sec^2a-1. Showing work is not bad but we are trying to be elegant. Have I overused that word yet? Where was I...oh yeah. Elegance however does not mean skipping steps! Work will need to be shown if the formulas above aren't present in the question. Another thing about the picture above, it is not needed to memorize all those formulas. They are all derived from sin^2x+cos^2x=1. That is the only important one to remember.
This lesson also focused on do's and dont's when solving an identity. The first thing we learned was the "Great Wall of China", which is the line we put under the equal sign. This line can't be passed! This is because it is not an equation. We can't do things like multiplying or adding to balance both sides. However we can "algaebraically massage" the seperate sides into something we can work with. Another don't is putting equal signs, instead we just show the work going down. Putting an equal sign is a no no because we don't know till the end of the work if one side is really equal to the other side. We were also shown some strategies to solving identities, which are above.
So we started the morning class off by watching that article 13 video. After that it was straight to math. Today's math class was a workshop since many of us enjoy doing this activity.
Anyways, the lesson for today had a great deal of emphasis on...elegance in showing our work. By elegance, I mean the quickest and most direct way of solving or proofing an identity. The formulas above help a great deal in achieving this elegance. To make things elegant we can change everything into sine, cosine, or any of the other identies to solve faster. Like the question on slide 3. Work did not need to be shown because it is already proven that tan^2a is = sec^2a-1. Showing work is not bad but we are trying to be elegant. Have I overused that word yet? Where was I...oh yeah. Elegance however does not mean skipping steps! Work will need to be shown if the formulas above aren't present in the question. Another thing about the picture above, it is not needed to memorize all those formulas. They are all derived from sin^2x+cos^2x=1. That is the only important one to remember.
This lesson also focused on do's and dont's when solving an identity. The first thing we learned was the "Great Wall of China", which is the line we put under the equal sign. This line can't be passed! This is because it is not an equation. We can't do things like multiplying or adding to balance both sides. However we can "algaebraically massage" the seperate sides into something we can work with. Another don't is putting equal signs, instead we just show the work going down. Putting an equal sign is a no no because we don't know till the end of the work if one side is really equal to the other side. We were also shown some strategies to solving identities, which are above.
Then we were given questions like these, where we could apply those strategies. For the first question, we multiplied top and bottom by 1+sint because we can see 1-sint is a factor of a difference of squares. We multiply it by its other factor to make things easier. Now the top is (1+sint)cos and the bottom (1-sin^2t). The bottom can be changed to a simpler form which is cos^2t! We know that sin^2t + cos^2t=1 without proving it. So after moving sin^2t to the other side we'll get cos^2t=1-sin^2t. That is elegance folks! Then we just continue simplifying it as much as possible. When we're all done we put Q.E.D to indicate we are finished. Q.E.D is some latin word, which I can't remember. So don't ask XD.
The rest of the questions we're mostly the same but using different strategies like factoring and such. Doing these questions need a great deal of practice to get better! Alrightey, there's my scribe post. I hope I included all the important things so yeah. I shall see everyone tmr. Bye bye.
Whoops forgot to name the next scribe. You know who you are! AnhThi. Also, why won't my paragraphs seperate up there...oh well.
Monday, March 17, 2008
Article 13, Al Upton, and the minilegends
If you'd like to leave a comment to Al Upton and the minilegends in Australia click on the picture below to get to their blog.
You can also read more about Article 13 and the Convention on the Rights of the Child here. This movie illustrates what it's all about:
You can also read more about Article 13 and the Convention on the Rights of the Child here. This movie illustrates what it's all about:
Pythagorean Identities
OUTLINE:
* Discussed a case study, a blogging issue, that is of recent concern in Australia.
* Continued on solving the introductory expressions on trigonometric identites.
* Discovered the concept of corrolories and how it they can make 'algebraically massaging' an expression much easier.
CASE STUDY: AL UPTON AND THE MINILEGENDS ARE SHUT DOWN
The day started off with an appealing discussion on a man named Mr. Al Upton, a sixth grade teacher in Australia, whose blog for his students was recently shut down by order of the Ministry of Education. The reasons behind the shutting down of his educational blog is not specific, but this decision is believed to contradict Article 13 of the Convention on the Rights of the Child in which the child is allowed to use any type of media for her or his education, basically. This alarming issue was discussed by Mr. Kuropatwa to his class as a warning that our blogs could be shut down as well. Mr. K encouraged us to comment on Upton's blog so as to raise the concern even further that Canadian children (us) are concerned about this as well and that children should be able to use blogs for educational purposes.
Now on to today's lesson.
INTRODUCTORY EXPRESSIONS OF TRIGONOMETRIC IDENTITIES
Refer to SLIDES 4 to 7.
We 'algebraically massaged' (sec t)/(cos t) - (sec t)(cos t) and (1 + tan^2 B)/ tan^2 B. But that work is now lost and Mr.K had to recreate them as that is mentioned in his post.
PYTHAGOREAN IDENTITIES
Refer to SLIDE 8.
One can find that any triangle on the circle that the equation...
sin^2 X + cos^2 X = 1
...with the case of degenerate triangles at the corner points on the unit circle (i.e., 0, pi/2, pi, 3pi/2, and 2pi in the domain of [0,2pi]) in which the 'triangle' has an adjacent or opposite side equal to zero, therefore, "missing a hypotenuse" and the 'triangle' can be seen as "degenerate," but still abides to the "sin^2 X + cos^2 X = 1" rule.
DERIVING COROLLARIES FROM sin^2 X + cos^2 X = 1.
Refer to SLIDE 9.
If sin^2 X + cos^2 X = 1, then the following must also be true...
sin^2 X = 1 - cos^2 X
cos^2 X = 1 - sin^2 X
...because they are algebraically the same.
So anytime we can see sin^2 X, cos^2 X, 1 - cos^2 X, and 1 - sin^2 X in doing trigonometric identities questions, we can substitute those in.
For example, refer to SLIDE 10.
From our previous class, we 'algebraically massaged' (1 - cos^2 A)/cos^2 A. We substituted/replaced the 1 - cos^2 A with a sin^2 A and found out that the equation can further be simplified to (sin^2 A)/(cos^2 A) or tan^2 A.
Refer to SLIDE 11.
And if we use that same equation, sin^2 X + cos^2 X = 1, and divided that whole equation by sin^2 X or cos^2 X, we can derive more corrolaries that are algebraically the same as its siblings.
SUMMARY
* Mr. Al Upton and the miniLegends' blog was shut down by the Ministry of Education in Australia because the ministry felt the blog violated some laws. This can happen to us!
* From sin^2 X + cos^2 X = 1, we can derive many more equations--the corrolaries--and use them in 'algebraically massaging' trigonometric identities questions.
HOMEWORK:
Exercise 14: Trigonometric Identities I
Next scribe is Joyce.
* Discussed a case study, a blogging issue, that is of recent concern in Australia.
* Continued on solving the introductory expressions on trigonometric identites.
* Discovered the concept of corrolories and how it they can make 'algebraically massaging' an expression much easier.
CASE STUDY: AL UPTON AND THE MINILEGENDS ARE SHUT DOWN
The day started off with an appealing discussion on a man named Mr. Al Upton, a sixth grade teacher in Australia, whose blog for his students was recently shut down by order of the Ministry of Education. The reasons behind the shutting down of his educational blog is not specific, but this decision is believed to contradict Article 13 of the Convention on the Rights of the Child in which the child is allowed to use any type of media for her or his education, basically. This alarming issue was discussed by Mr. Kuropatwa to his class as a warning that our blogs could be shut down as well. Mr. K encouraged us to comment on Upton's blog so as to raise the concern even further that Canadian children (us) are concerned about this as well and that children should be able to use blogs for educational purposes.
Now on to today's lesson.
INTRODUCTORY EXPRESSIONS OF TRIGONOMETRIC IDENTITIES
Refer to SLIDES 4 to 7.
We 'algebraically massaged' (sec t)/(cos t) - (sec t)(cos t) and (1 + tan^2 B)/ tan^2 B. But that work is now lost and Mr.K had to recreate them as that is mentioned in his post.
PYTHAGOREAN IDENTITIES
Refer to SLIDE 8.
One can find that any triangle on the circle that the equation...
sin^2 X + cos^2 X = 1
...with the case of degenerate triangles at the corner points on the unit circle (i.e., 0, pi/2, pi, 3pi/2, and 2pi in the domain of [0,2pi]) in which the 'triangle' has an adjacent or opposite side equal to zero, therefore, "missing a hypotenuse" and the 'triangle' can be seen as "degenerate," but still abides to the "sin^2 X + cos^2 X = 1" rule.
DERIVING COROLLARIES FROM sin^2 X + cos^2 X = 1.
Refer to SLIDE 9.
If sin^2 X + cos^2 X = 1, then the following must also be true...
sin^2 X = 1 - cos^2 X
cos^2 X = 1 - sin^2 X
...because they are algebraically the same.
So anytime we can see sin^2 X, cos^2 X, 1 - cos^2 X, and 1 - sin^2 X in doing trigonometric identities questions, we can substitute those in.
For example, refer to SLIDE 10.
From our previous class, we 'algebraically massaged' (1 - cos^2 A)/cos^2 A. We substituted/replaced the 1 - cos^2 A with a sin^2 A and found out that the equation can further be simplified to (sin^2 A)/(cos^2 A) or tan^2 A.
Refer to SLIDE 11.
And if we use that same equation, sin^2 X + cos^2 X = 1, and divided that whole equation by sin^2 X or cos^2 X, we can derive more corrolaries that are algebraically the same as its siblings.
SUMMARY
* Mr. Al Upton and the miniLegends' blog was shut down by the Ministry of Education in Australia because the ministry felt the blog violated some laws. This can happen to us!
* From sin^2 X + cos^2 X = 1, we can derive many more equations--the corrolaries--and use them in 'algebraically massaging' trigonometric identities questions.
HOMEWORK:
Exercise 14: Trigonometric Identities I
Next scribe is Joyce.
Saturday, March 15, 2008
Identities
Hello everyone, it's Roxanne blogging for Friday's class, March 14, aka PI DAY! Anyways.. I know it's pretty late but might as well complete my scribe now before I forget during the weekend.
In the morning we started off by celebrating Pi Day! WOOHOO! We had some pie, listened to the Pi song by Kate Bush, and also answered some trivia questions. While eating some pie, we began our new unit on Identities. What we must remember is that, there are many ways to express and equation. They may look very different but by algebraically massaging the equation, you can get the same value of the original equation but it will look much different. The basic building blocks of all trigonometric functions are sine and cosince functions. In order to become very good at simplifying these equations, you need a lot of practice. Therefore, we will be practicing on figuring out how to complete these equations in future classes.
On the second slide, it shows the same face but different hair styles. However, it is still the same person but different hair.
On the 4th, 5th, and 6th slide we worked on various questions to find out each solution. And remember to keep your answer simple by simplifying.
In the afternoon class, we had some more PIE (: yay! But we also wrote our test for the unit Transformations. Truth be told I found a few questions confusing but hopefully everyone else did very well! Other than that, I believe I've completed the first scribe for the unit on Trigonometric Identities. Next scribe well be.........Joseph! (:
In the morning we started off by celebrating Pi Day! WOOHOO! We had some pie, listened to the Pi song by Kate Bush, and also answered some trivia questions. While eating some pie, we began our new unit on Identities. What we must remember is that, there are many ways to express and equation. They may look very different but by algebraically massaging the equation, you can get the same value of the original equation but it will look much different. The basic building blocks of all trigonometric functions are sine and cosince functions. In order to become very good at simplifying these equations, you need a lot of practice. Therefore, we will be practicing on figuring out how to complete these equations in future classes.
On the second slide, it shows the same face but different hair styles. However, it is still the same person but different hair.
On the 4th, 5th, and 6th slide we worked on various questions to find out each solution. And remember to keep your answer simple by simplifying.
In the afternoon class, we had some more PIE (: yay! But we also wrote our test for the unit Transformations. Truth be told I found a few questions confusing but hopefully everyone else did very well! Other than that, I believe I've completed the first scribe for the unit on Trigonometric Identities. Next scribe well be.........Joseph! (:
Friday, March 14, 2008
Thursday, March 13, 2008
BOBbing for.... knowledge
Hello, this is Paul, posting his Blogging on Blogging post.
For this unit, Transformations, we covered a wide variety of topics that relate to graphing. Most of these were pretty simple, like DABC and flips, shifts and stretches. However, for me atleast, I find the questions involving inverses to be confusing most of the time despite how the answer is fairly simple. Also, Im still not really getting how to graph the 1/f(x) questions, although I understand what is being said I can't seem to apply it myself. The question we did in place of the test also helped me alot in understanding how to solve "asin[b(x-c)]+d" questions, which I did not fully know how to solve before. Now I can solve one with a calculator and some paper, yay. Thanks Mr.K.
So in summary, difficult things are:
Graphing 1/f(x) functions. I think my problem here is that Im still looking at the graph as points and then trying to apply the "1/f(x)" (which was our approach for graphing "asin[b(x-c)]+d"), which means I end up with points but not a proper graph. I should definitely review the scribe post for that class before the test.
Inverses are just a little confusing, but I do understand what they do and mean. Probably the hardest part for me is graphing one because Im not totally sure about how to, but a look over that scribe post will probably help.
Other than that I think I understand this unit, or I understand well enough to pass this test (hopefully).
I also would like to apologize to Lawrence and Nelsa that I did not post our group's solution in time. I unfortunately did not anticipate that my alarm clock would not wake me up. Due to my alarm clock being off, I did not wake up until nearly 3pm. Much thanks to Lawrence for posting our solution.
So good luck guys, and don't forget to bring delicious pie!
For this unit, Transformations, we covered a wide variety of topics that relate to graphing. Most of these were pretty simple, like DABC and flips, shifts and stretches. However, for me atleast, I find the questions involving inverses to be confusing most of the time despite how the answer is fairly simple. Also, Im still not really getting how to graph the 1/f(x) questions, although I understand what is being said I can't seem to apply it myself. The question we did in place of the test also helped me alot in understanding how to solve "asin[b(x-c)]+d" questions, which I did not fully know how to solve before. Now I can solve one with a calculator and some paper, yay. Thanks Mr.K.
So in summary, difficult things are:
Graphing 1/f(x) functions. I think my problem here is that Im still looking at the graph as points and then trying to apply the "1/f(x)" (which was our approach for graphing "asin[b(x-c)]+d"), which means I end up with points but not a proper graph. I should definitely review the scribe post for that class before the test.
Inverses are just a little confusing, but I do understand what they do and mean. Probably the hardest part for me is graphing one because Im not totally sure about how to, but a look over that scribe post will probably help.
Other than that I think I understand this unit, or I understand well enough to pass this test (hopefully).
I also would like to apologize to Lawrence and Nelsa that I did not post our group's solution in time. I unfortunately did not anticipate that my alarm clock would not wake me up. Due to my alarm clock being off, I did not wake up until nearly 3pm. Much thanks to Lawrence for posting our solution.
So good luck guys, and don't forget to bring delicious pie!
Group Second to None
Okay, late I know, but unfortunately our group didn't post in time, So I'll try to do this quick fast.
Anyways, this is our solution.
SLIDE 17:
This equation gives the depth of the water, h meters, at an ocean port at any time, t hours during a certain day
h(t) = 2.5 sin[2pi(t - 1.5)/12.4] + 4.3
A) Explain the significance of each number in the equation
II) 12.4 is the Period of the function, which is obtained from parameter B - 2pi/12.4
III) 1.5 - This is parameter C, which is the phase shift, in which this function, the graph shifts to the right 1.5 units/hours.
IV)4.3 - This is parameter D, which is the sinusoidal axis, which shifts the sinusoidal axis up 4.3 metres..
B) What is the minimum depth of the water? When does it occur?
Now, We can go backwards and use that point, but since we can use whichever point, we decided to use the next one, which minimum depth, 1.8 metres, occurs at 10.8 hours.
C) Determine the depth of the water at 9:30 am.
So then we just plug it in.
h(9.5) = 2.5sin[2pi(9.5 - 1.5)/12.4] + 4.3
*t is 9.5 because time ~ 9:30 AM ~ is converted to 9.5 because :30 minutes is .5 hours.*
in which we get... 2.323 metres.
D) Determine one time when the water is 4.0 metres deep.
So then we go something like...
4 = 2.5sin[2pi(t-1.5)/12.4] + 4.3
First we subtracted 4.3 from both sides like so.
-0.3 = 2.5sin[2pi(t-1.5)/12.4]
Then we added the phase shift to both sides to get...
1.2 = 2.5 sin[2pi(t)/12.4]
Then we divided parameter A out.
0.48 = sin[2pi(t)/12.4)
We then divided 2pi/12.4 in which .48 would be multiplied by the reciprocal of 2pi/12.4, and moved sine to the other side to change it to ARCSine to isolate t.
ARCSine(0.9473) = t
Which would equal to --> 1.2447 hours.
To make it efficient, we'll multiply the .2247 by 60 so
Depth of water of 4 metres occurs at 1:13:48 am.
Again, sorry for posting so late. Original personnel that was to post, did not post. Feel free to comment, as we are supposed to.
Anyways, this is our solution.
SLIDE 17:
This equation gives the depth of the water, h meters, at an ocean port at any time, t hours during a certain day
h(t) = 2.5 sin[2pi(t - 1.5)/12.4] + 4.3
A) Explain the significance of each number in the equation
II) 12.4 is the Period of the function, which is obtained from parameter B - 2pi/12.4
III) 1.5 - This is parameter C, which is the phase shift, in which this function, the graph shifts to the right 1.5 units/hours.
IV)4.3 - This is parameter D, which is the sinusoidal axis, which shifts the sinusoidal axis up 4.3 metres..
B) What is the minimum depth of the water? When does it occur?
Now, We can go backwards and use that point, but since we can use whichever point, we decided to use the next one, which minimum depth, 1.8 metres, occurs at 10.8 hours.
C) Determine the depth of the water at 9:30 am.
So then we just plug it in.
h(9.5) = 2.5sin[2pi(9.5 - 1.5)/12.4] + 4.3
*t is 9.5 because time ~ 9:30 AM ~ is converted to 9.5 because :30 minutes is .5 hours.*
in which we get... 2.323 metres.
D) Determine one time when the water is 4.0 metres deep.
So then we go something like...
4 = 2.5sin[2pi(t-1.5)/12.4] + 4.3
First we subtracted 4.3 from both sides like so.
-0.3 = 2.5sin[2pi(t-1.5)/12.4]
Then we added the phase shift to both sides to get...
1.2 = 2.5 sin[2pi(t)/12.4]
Then we divided parameter A out.
0.48 = sin[2pi(t)/12.4)
We then divided 2pi/12.4 in which .48 would be multiplied by the reciprocal of 2pi/12.4, and moved sine to the other side to change it to ARCSine to isolate t.
ARCSine(0.9473) = t
Which would equal to --> 1.2447 hours.
To make it efficient, we'll multiply the .2247 by 60 so
Depth of water of 4 metres occurs at 1:13:48 am.
Again, sorry for posting so late. Original personnel that was to post, did not post. Feel free to comment, as we are supposed to.
Nothing Less Than The Best....Group
SLIDE 18
On a typical day at an ocean port, the water has a maximum depth of 20 m at 8:00 a.m. The minimum depth of 8 m occurs 6.2 hours later. Assume that the relation between the depth of the water and time is a sinusoidal function.
Let's draw a graph!
a) What is the period of the function?
From the information we have been given...
* We can set the 8am as t = 0 hours.
* A maximum value is when t = 0 hours and when d = 20.
* A minimum value is when t = 6.2 and d = 8.
We can see from the graph that the period is 12.4 hours.
b) Write an equation for the depth of the water at any time, t hours.
cosine equation's parameters...
A = 6
B = (2pi) / 12.4 = pi/6.2
C = 0
D = 14
To get A, amplitude, calculate the distance from the sinusoidal axis to a maximum value or minimum value.
sinusoidal axis = (20+8)/2 = 14
amplitude = 14-8 = 6
B = pi/period = pi/6.2
C, the phase shift, is 0.
D is the sinusoidal axis, 14.
D(t) = 6cos [(pi/6.2)t] + 14
c) Determine the depth of the water at 10:00 a.m.
10:00 am = 2 hrs from when t = 0 or 8:00 am. Plug in the 2 as t into the equation to get the answer.
D(2) = 6 cos [(pi/6.4)2] + 14 = 17.1738 metres
d) Determine one time when the water is 10 m deep.
The wave is 10 metres deep, so the qestion is asking for what the time is when D = 10. Plug in 10 as D, then solve for t.
10 = 6cos[(pi/6.2)t] +14
-4 = 6cos[(pi/6.2)t]
-4/6 = cos[(pi/6.2)t]
arc cos(-4/6) = (pi/6.2)t
2.3005 = (pi/6.2)t
2.3005/(pi/6.2) = t
2.3005 x 6.2/pi = t
14.2632/pi = t
t = 4.5401
Convert the 4.5401 into "actual time" because we use hours:minutes:seconds to show time, so...
4.5401 hrs + 8am = 12.5401 hrs
Obviously, its not efficient to say .5401 hrs so we convert that to minutes.
0.5401 x 60 = 32.406 min.
12:32:24pm
We can round that to 12:30pm.
On a typical day at an ocean port, the water has a maximum depth of 20 m at 8:00 a.m. The minimum depth of 8 m occurs 6.2 hours later. Assume that the relation between the depth of the water and time is a sinusoidal function.
Let's draw a graph!
a) What is the period of the function?
From the information we have been given...
* We can set the 8am as t = 0 hours.
* A maximum value is when t = 0 hours and when d = 20.
* A minimum value is when t = 6.2 and d = 8.
We can see from the graph that the period is 12.4 hours.
b) Write an equation for the depth of the water at any time, t hours.
cosine equation's parameters...
A = 6
B = (2pi) / 12.4 = pi/6.2
C = 0
D = 14
To get A, amplitude, calculate the distance from the sinusoidal axis to a maximum value or minimum value.
sinusoidal axis = (20+8)/2 = 14
amplitude = 14-8 = 6
B = pi/period = pi/6.2
C, the phase shift, is 0.
D is the sinusoidal axis, 14.
D(t) = 6cos [(pi/6.2)t] + 14
c) Determine the depth of the water at 10:00 a.m.
10:00 am = 2 hrs from when t = 0 or 8:00 am. Plug in the 2 as t into the equation to get the answer.
D(2) = 6 cos [(pi/6.4)2] + 14 = 17.1738 metres
d) Determine one time when the water is 10 m deep.
The wave is 10 metres deep, so the qestion is asking for what the time is when D = 10. Plug in 10 as D, then solve for t.
10 = 6cos[(pi/6.2)t] +14
-4 = 6cos[(pi/6.2)t]
-4/6 = cos[(pi/6.2)t]
arc cos(-4/6) = (pi/6.2)t
2.3005 = (pi/6.2)t
2.3005/(pi/6.2) = t
2.3005 x 6.2/pi = t
14.2632/pi = t
t = 4.5401
Convert the 4.5401 into "actual time" because we use hours:minutes:seconds to show time, so...
4.5401 hrs + 8am = 12.5401 hrs
Obviously, its not efficient to say .5401 hrs so we convert that to minutes.
0.5401 x 60 = 32.406 min.
12:32:24pm
We can round that to 12:30pm.
Trig Assignment team "Jabawakeez"?
So these are the solutions for the question:
Here's how our graph looks like. In the question we were given the max and min values of the wave. The max is at 4:30 and we changed the time to its number value by taking the minutes and dividing it by 60 so its new value was 4.5 on the graph. Now it is easier to add 6.2 hours to it, so the min will be at 10.7 on the x axis. With the max and min we can figure out 1/2 of the period. Since the min and max are 6.2 hours apart we use that. 6.2 * 2 = 12.4. But here's the tricky part. If you do 12.4*(1/4) it will not equal 4.5. So we can't do the 1,2,3,4 ticks and automatically write in the values. What we did was 4.5+12.4 to make it easier to graph. The value in the 4th tick on the x axis is now 16.9. The values on the x axis are now 1/4, 1/2, 3/4 of 16.9. Then you'll just put the points on at the max and min. Then we know the patterns of the curve (after the max) will go to the avg value, then min, avg value again, and then the max.
a)Here's an equation in cose. To find D we found the average of the min and max which was 5. We can now find parameter a by subtracting 5 from the max 9.6. B=2pi/period so in our equation it'll be 2pi/12. In our graph here it does not start at the max where cosine starts so it must've moved by 4.5 our parameter c.
b) First we changed 2:46pm to its number value. Since it has passed into night time we can add the 12 hours from the morning. Then all we do is add the 2 hours and 46 minutes. But we divide 46/60 first. So its number value should be 14.7667. We take this value and plug it into t. This can be done on the calculator...and will get an answer of h=7.1641m.
c)In this question we just plugged in 2 into the equation and solved for t.
Tidal forces are greatest when Earth, the sun, and the moon are in line. When this occurs at the Annapolis Tidal Generating Station, the water has a maximum depth of 9.6 m at 4:30 am and a minimum depth of 0.4m 6.2 hours later.
Here's how our graph looks like. In the question we were given the max and min values of the wave. The max is at 4:30 and we changed the time to its number value by taking the minutes and dividing it by 60 so its new value was 4.5 on the graph. Now it is easier to add 6.2 hours to it, so the min will be at 10.7 on the x axis. With the max and min we can figure out 1/2 of the period. Since the min and max are 6.2 hours apart we use that. 6.2 * 2 = 12.4. But here's the tricky part. If you do 12.4*(1/4) it will not equal 4.5. So we can't do the 1,2,3,4 ticks and automatically write in the values. What we did was 4.5+12.4 to make it easier to graph. The value in the 4th tick on the x axis is now 16.9. The values on the x axis are now 1/4, 1/2, 3/4 of 16.9. Then you'll just put the points on at the max and min. Then we know the patterns of the curve (after the max) will go to the avg value, then min, avg value again, and then the max.
a)Here's an equation in cose. To find D we found the average of the min and max which was 5. We can now find parameter a by subtracting 5 from the max 9.6. B=2pi/period so in our equation it'll be 2pi/12. In our graph here it does not start at the max where cosine starts so it must've moved by 4.5 our parameter c.b) First we changed 2:46pm to its number value. Since it has passed into night time we can add the 12 hours from the morning. Then all we do is add the 2 hours and 46 minutes. But we divide 46/60 first. So its number value should be 14.7667. We take this value and plug it into t. This can be done on the calculator...and will get an answer of h=7.1641m.
c)In this question we just plugged in 2 into the equation and solved for t.
Wednesday, March 12, 2008
WORD PROBLEMS...TEAM KA-BLAMO! Problems?? Word, man...
I think it's original to call our fantabulous team of three Team Ka-Blamo [members: Jamie, Kristina and Eleven]
Well, our problem of course, was on slide 15 and it asks:
Well, we're going to do things the old school modern way...actually write the stuff on paper...[because we can't afford a tablet] then scan it and upload...that's the way we do things. We spent like five hours getting everything well, thorough enough via MSN...
A FERRIS WHEEL HAS A RADIUS OF 20 METERS. IT ROTATES ONCE EVERY 40 SECONDS. PASSENGERS GET ON AT POINT S WHICH IS 1 METERS ABOVE GROUND LEVEL. SUPPOSE YOU GET ON AT S AND THE WHEEL STARTS TO ROTATE.
a.] Graph how your height above the ground varies during the first two cycles...
The graph we drew looks pointy but it is CURVED sorry.
That's just a graph based on the given info.... and we nearly got confused because of the minimum value being 1 m off the ground instead of touching the x-axis. But then we figured that ferris WHEELS [typo in your slide.] don't touch the ground or else they would scrape the concrete.
b.] Write an equation that expresses your height as a function of the elapsed time.
Basically, all we did was find each parameter for both the sine and cosine function even though we only needed one.
c.] Determine your height above the ground after 45 seconds.
All that needs to be done here is sub the t in the function H(t) with 45 seconds and the result will be the new height at this time according to the graph, assuming the revolving doesn't stop and is continuous.
d] Determine one time when your height is 35m above the ground.
Replace H(t) with 35 and isolate t to get time in seconds.
That wasn't much explanation in words but you know... we spent a lot of time getting these answers and we hope this is what you guys are looking for....I'm famished. G'night...I'll edit this later....haha
Well, our problem of course, was on slide 15 and it asks:
Well, we're going to do things the old school modern way...actually write the stuff on paper...[because we can't afford a tablet] then scan it and upload...that's the way we do things. We spent like five hours getting everything well, thorough enough via MSN...
A FERRIS WHEEL HAS A RADIUS OF 20 METERS. IT ROTATES ONCE EVERY 40 SECONDS. PASSENGERS GET ON AT POINT S WHICH IS 1 METERS ABOVE GROUND LEVEL. SUPPOSE YOU GET ON AT S AND THE WHEEL STARTS TO ROTATE.
a.] Graph how your height above the ground varies during the first two cycles...
The graph we drew looks pointy but it is CURVED sorry.
That's just a graph based on the given info.... and we nearly got confused because of the minimum value being 1 m off the ground instead of touching the x-axis. But then we figured that ferris WHEELS [typo in your slide.] don't touch the ground or else they would scrape the concrete.
b.] Write an equation that expresses your height as a function of the elapsed time.
Basically, all we did was find each parameter for both the sine and cosine function even though we only needed one.
c.] Determine your height above the ground after 45 seconds.
All that needs to be done here is sub the t in the function H(t) with 45 seconds and the result will be the new height at this time according to the graph, assuming the revolving doesn't stop and is continuous.
d] Determine one time when your height is 35m above the ground.
Replace H(t) with 35 and isolate t to get time in seconds.
That wasn't much explanation in words but you know... we spent a lot of time getting these answers and we hope this is what you guys are looking for....I'm famished. G'night...I'll edit this later....haha
Yet Another Workshop Class
Hello once again, it is I, Kristina, back for another scribe post. Today, we were supposed to have a pre-test in the morning and the actual test in the afternoon but we ended up only having the pre-test in the afternoon. The reason why was because Mr. K thought that we weren't ready to have the test after seeing us somewhat struggle over yesterday's class. So, in the end, we ended up using the morning class to have a workshop period! Yay, fun..
After we had been split into our groups, which there were a total of four, we were given a question to work with. I will type out the question, which is already on slide 2, for your convenience.
At a sea port, the depth of the water, h meters, at time, t hours, during a certain day is given this formula:
A) State the: i) period ii) amplitude iii) phase shift
This was the first part of the question, which should be fairly straight forward. BUT! It ended up not being so straight forward to the class. Every group came to the same conclusion for the amplitude and phase shift, but the class became divided with finding the period. Which is shown in slide 2 with the two different answers.
The first answer (red) is not the right answer, the one in blue is the correct one. This is because the group who put up answer red didn't factor out parameter B correctly. They thought that because since 2pi wasn't "visually" over 12.4, that they had to factor that out first and then multiply 2pi by 12.4 to isolate parameter C. After a bit of arguing and confusion, Mr. K kindly showed us that even though the 2pi didn't appear to be over 12.4, it actually was. The reason for this is because that 2pi could also be seen as 2pi/1, which, when multiplied, actually gives 2pi(t-4.00)/12.4! Now, from there all you need to do to get the period is to factor out the 2pi/12.4 to get parameter B and then to get divide that from 2pi. Voila, you should then get the answer of 12.4. We can then move onto the next question..
B) What is the maximum depth of the water? When does it occur?
Now, for this part of the question, I am going to explain how to do it in a completely different way from what the slide shows you. The reason for this is because, in my own opinion, it doesn't depict a very accurate and understandable way on how to find the actual value. By just using the graph to try and find it, I think its just like guessing, therefore not very accurate. The graph is only good when you want to pinpoint which areas the answer will be in, which is actually shown very nicely on the graph in slide 4.
Well then, to show how to get 7.1 hrs as the answer, I am going to show you how to get it MATHEMATICALLY using the formula!
Looks difficult? Nah! Its so simple! You see, all I did was input the max value, which was 4.9 into h(t) . From there, I then solved for t. I'm sure most of you have gotten stuck after getting to the ARCsin part, well then don't fret, because I've also had troubles with that in the past. All you have to do to isolate the (t-4) is to multiply both sides by the reciprocal of 12.4/2pi. When you do this, 2pi/12.4 on the right side will get cancelled out and then you will be left with 12.4/2pi * ARCsin(1) = t - 4. From there's its baby stuff! We're all in pre cal, I don't think you need me to explain what to do from there. Anyways, I hope this solves any problems people had with slide 4. MOVING ON..
C) Determine the depth of water at 5:00 A.M. and at 12:00 noon.
This is simple, all you have to do is input 5 and 12 into the value for t and then solve from there. If you need further explanation for this, feel free to add a comment since I am seriously getting tired right now haha >_<.
D) Determine one time when the water is 2.25 meters deep.
Okay, this is basically the same as letter B. Instead of 4.9, input 2.25. Then solve from there. No further explanations need to be said for how to do these kind of questions. But, in case you do, please leave a comment!
AWWRIIIGHT! Moving onto the afternoon class! First thing we did was have a PRE-TEST! We were given about, I don't know, 20 minutes? to do it and then we were put back into the same groups we were in for the morning class to discuss our answers so that we could hand it in, but only one person's pre-test was handed in.
We then went over the answers after having handed in one person's pre-test. The multiple choice answers were pretty straightforward, so I am going to skip to the very last question since it was definitely the hardest one out of them all.
Starting off with the question "a)" of the last question. Its asked to sketch a graph of the height of the point A above the outflow water level as a function of time starting at t = 0 seconds, with A as shown in the diagram. The first thing you had to do to be able to graph this was to find the period. We were able to get the period from the information in the question that said there were 5 revolutions every 4 minutes. From this info, we can then find the period by attempting to when 1 revolution occurs, which will end up in seconds. As slide 12 shows, the period will be 48 seconds since that is when 1 revolution occurs. The graph can now be made easily and the reason why it starts at 0 is because starting at 0 seconds, point A is still above the outflow.
Part "b)" of the question is pretty self explanatory. All you had to do was take a look at the graph and then find the different parameters for SINE and COSINE. From there, you would be able to get the same answers as shown in the slide if you did it right. As for part "c)", this is just like question "B" from the morning class. Instead, you can choose between which formula you want to work with, either the SINE or the COSINE that you made in the previous question. Once you've done that, just input 4 into h(t) and then solve for "t" in the exact same way as shown in "B" from the morning's question. After having solved for question "c)", you can now solve for question "d)". Take a quick look back at the graph and find where 4 meters would be. If you draw a straight line across, you'd see that it touches two parts of the wave. That is why the answer shows that you have to multiply the answer by 2.
After we had finished going over all the answers, Mr. K then told us that we had a group assignment to do. Each group was given an assignment that was similar to the questions we had done today. They're expected to be posted onto the blog with the appropriate tags: "TrigAssignment, (names of group members), Transformation. The due date is on MARCH 13 BY NOON. Once every group has got their assignment posted, everyone is expected to comment on each of the groups' assignments, excluding their own, whether they be about how neat their work was, or if they found an error, etc.
Now then, I guess that's all for me. REMEMBER GUYS! TOMORROW'S PI DAY! Make sure you don't forget to bring your delicious pies! As for the lucky scribe for Pi Day, I choose....Roxanne. Well then, that's all for me. Oh yeah, and remember! THE TRANSFORMATIONS TEST IS ON FRIDAY AFTERNOON! For those of you who haven't BOB'd yet, be sure to do it by then! Alrighty, now seriously, I'm out.
After we had been split into our groups, which there were a total of four, we were given a question to work with. I will type out the question, which is already on slide 2, for your convenience.
At a sea port, the depth of the water, h meters, at time, t hours, during a certain day is given this formula:
A) State the: i) period ii) amplitude iii) phase shift
This was the first part of the question, which should be fairly straight forward. BUT! It ended up not being so straight forward to the class. Every group came to the same conclusion for the amplitude and phase shift, but the class became divided with finding the period. Which is shown in slide 2 with the two different answers.
The first answer (red) is not the right answer, the one in blue is the correct one. This is because the group who put up answer red didn't factor out parameter B correctly. They thought that because since 2pi wasn't "visually" over 12.4, that they had to factor that out first and then multiply 2pi by 12.4 to isolate parameter C. After a bit of arguing and confusion, Mr. K kindly showed us that even though the 2pi didn't appear to be over 12.4, it actually was. The reason for this is because that 2pi could also be seen as 2pi/1, which, when multiplied, actually gives 2pi(t-4.00)/12.4! Now, from there all you need to do to get the period is to factor out the 2pi/12.4 to get parameter B and then to get divide that from 2pi. Voila, you should then get the answer of 12.4. We can then move onto the next question..
B) What is the maximum depth of the water? When does it occur?
Now, for this part of the question, I am going to explain how to do it in a completely different way from what the slide shows you. The reason for this is because, in my own opinion, it doesn't depict a very accurate and understandable way on how to find the actual value. By just using the graph to try and find it, I think its just like guessing, therefore not very accurate. The graph is only good when you want to pinpoint which areas the answer will be in, which is actually shown very nicely on the graph in slide 4.
Well then, to show how to get 7.1 hrs as the answer, I am going to show you how to get it MATHEMATICALLY using the formula!
Looks difficult? Nah! Its so simple! You see, all I did was input the max value, which was 4.9 into h(t) . From there, I then solved for t. I'm sure most of you have gotten stuck after getting to the ARCsin part, well then don't fret, because I've also had troubles with that in the past. All you have to do to isolate the (t-4) is to multiply both sides by the reciprocal of 12.4/2pi. When you do this, 2pi/12.4 on the right side will get cancelled out and then you will be left with 12.4/2pi * ARCsin(1) = t - 4. From there's its baby stuff! We're all in pre cal, I don't think you need me to explain what to do from there. Anyways, I hope this solves any problems people had with slide 4. MOVING ON..C) Determine the depth of water at 5:00 A.M. and at 12:00 noon.
This is simple, all you have to do is input 5 and 12 into the value for t and then solve from there. If you need further explanation for this, feel free to add a comment since I am seriously getting tired right now haha >_<.
D) Determine one time when the water is 2.25 meters deep.
Okay, this is basically the same as letter B. Instead of 4.9, input 2.25. Then solve from there. No further explanations need to be said for how to do these kind of questions. But, in case you do, please leave a comment!
AWWRIIIGHT! Moving onto the afternoon class! First thing we did was have a PRE-TEST! We were given about, I don't know, 20 minutes? to do it and then we were put back into the same groups we were in for the morning class to discuss our answers so that we could hand it in, but only one person's pre-test was handed in.
We then went over the answers after having handed in one person's pre-test. The multiple choice answers were pretty straightforward, so I am going to skip to the very last question since it was definitely the hardest one out of them all.
Starting off with the question "a)" of the last question. Its asked to sketch a graph of the height of the point A above the outflow water level as a function of time starting at t = 0 seconds, with A as shown in the diagram. The first thing you had to do to be able to graph this was to find the period. We were able to get the period from the information in the question that said there were 5 revolutions every 4 minutes. From this info, we can then find the period by attempting to when 1 revolution occurs, which will end up in seconds. As slide 12 shows, the period will be 48 seconds since that is when 1 revolution occurs. The graph can now be made easily and the reason why it starts at 0 is because starting at 0 seconds, point A is still above the outflow.
Part "b)" of the question is pretty self explanatory. All you had to do was take a look at the graph and then find the different parameters for SINE and COSINE. From there, you would be able to get the same answers as shown in the slide if you did it right. As for part "c)", this is just like question "B" from the morning class. Instead, you can choose between which formula you want to work with, either the SINE or the COSINE that you made in the previous question. Once you've done that, just input 4 into h(t) and then solve for "t" in the exact same way as shown in "B" from the morning's question. After having solved for question "c)", you can now solve for question "d)". Take a quick look back at the graph and find where 4 meters would be. If you draw a straight line across, you'd see that it touches two parts of the wave. That is why the answer shows that you have to multiply the answer by 2.
After we had finished going over all the answers, Mr. K then told us that we had a group assignment to do. Each group was given an assignment that was similar to the questions we had done today. They're expected to be posted onto the blog with the appropriate tags: "TrigAssignment, (names of group members), Transformation. The due date is on MARCH 13 BY NOON. Once every group has got their assignment posted, everyone is expected to comment on each of the groups' assignments, excluding their own, whether they be about how neat their work was, or if they found an error, etc.
Now then, I guess that's all for me. REMEMBER GUYS! TOMORROW'S PI DAY! Make sure you don't forget to bring your delicious pies! As for the lucky scribe for Pi Day, I choose....Roxanne. Well then, that's all for me. Oh yeah, and remember! THE TRANSFORMATIONS TEST IS ON FRIDAY AFTERNOON! For those of you who haven't BOB'd yet, be sure to do it by then! Alrighty, now seriously, I'm out.
Tuesday, March 11, 2008
BOB on Transformations!
This is my first BOB before the test.
I found this unit particularly easy, that's ONLY if you were paying attention and reading your notes every night, like what Mr. K had explained you to do. Anyway. I was gone for a class, and every class of Mr. K's class missed hurts. You'll see if you were in my shoes. Especially if the lesson was long and essential. So I had some trouble on translations.
.. The .. y = a f(b(x-c)) + d
Everyone was going "STRETCHES BEFORE TRANSLATIONS"
And I was going "whuuuAAtt?" Or in double-click-other-words .. "What?!"
So I did my homework and went on the blog and caught up, or at least I believe I did.
I found transformations fair. I mean if you listen you should catch on quickly. Maybe a little more or less thinking here and there. The word problems aren't much of a biggie either. I'm confident of knowing my stuff. But not too confident in solving questions. Especially how there are many twists and turns of the question.
Oh right, and in the world problems. What I didn't like or did not understand was finding the time or day for a sunrise or certain time. It just got too confusing. Thanks.
I found this unit particularly easy, that's ONLY if you were paying attention and reading your notes every night, like what Mr. K had explained you to do. Anyway. I was gone for a class, and every class of Mr. K's class missed hurts. You'll see if you were in my shoes. Especially if the lesson was long and essential. So I had some trouble on translations.
.. The .. y = a f(b(x-c)) + d
Everyone was going "STRETCHES BEFORE TRANSLATIONS"
And I was going "whuuuAAtt?" Or in double-click-other-words .. "What?!"
So I did my homework and went on the blog and caught up, or at least I believe I did.
I found transformations fair. I mean if you listen you should catch on quickly. Maybe a little more or less thinking here and there. The word problems aren't much of a biggie either. I'm confident of knowing my stuff. But not too confident in solving questions. Especially how there are many twists and turns of the question.
Oh right, and in the world problems. What I didn't like or did not understand was finding the time or day for a sunrise or certain time. It just got too confusing. Thanks.
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