Alright .. Yesterday we had our Identities test.
-Oh sorry for the very very very very late blog.
Yes we had our test on identities. Personally I liked this one. Either I worked harder or the test was easier. Perhaps both. The test was pretty straight forward from what we were taught. Now to get to the main point!
The next scribe will NOT be Roxanne .. It will be .. RICHARD!
-#Eleven, out.
Showing posts with label Identities. Show all posts
Showing posts with label Identities. Show all posts
Wednesday, April 9, 2008
Tuesday, April 8, 2008
Hi my names BOB, and here about Identities...
So this is my BOB for our Identities unit.
At first, I didnt really like identities. The whole concept of "proving" instead of "solving" bothered me and I really didnt get it or like it at first. It took a little while and alot of practice before I was able to do it.
In general I think the identities stuff was pretty great, kinda easy. The hardest part was and still kinda is solving identities "elegantly." Its pretty confusing sometimes and I can never seem to remember some trig identities right. Since this was a really short unit and we didnt cover a wide array of different stuff, it was much easier to swallow, especially with all the practises we did. The sine dance helped alot, and was great fun.
I think im fairly well prepared for tomorrow, although I'll be sure to review notes before I go to bed and remind myself that cosa != a. Always making that mistake...
Anyway, good night everybody and good luck on the test!
At first, I didnt really like identities. The whole concept of "proving" instead of "solving" bothered me and I really didnt get it or like it at first. It took a little while and alot of practice before I was able to do it.
In general I think the identities stuff was pretty great, kinda easy. The hardest part was and still kinda is solving identities "elegantly." Its pretty confusing sometimes and I can never seem to remember some trig identities right. Since this was a really short unit and we didnt cover a wide array of different stuff, it was much easier to swallow, especially with all the practises we did. The sine dance helped alot, and was great fun.
I think im fairly well prepared for tomorrow, although I'll be sure to review notes before I go to bed and remind myself that cosa != a. Always making that mistake...
Anyway, good night everybody and good luck on the test!
Monday, April 7, 2008
Intro to Exponential Functions
Hi everybody this is benofschool and here is today's scribe.
Well we're back from spring break and everybody is ready to do some math! Just as a reminder everybody's Flickr pictures are due on the midnight of Friday. Just post a direct link to your Flickr page onto the blog. Remember to tag it with your name, flickr, and trigonometry. For the picture do not contrive it or draw it and say that it is math. It is math but it isn't naturally occuring. The picture must exist as it is naturally. One example of a contrived shot would be a drawing of a parabola on a piece of paper. It is drawn but a natural one would be on that I had taken last year:
My picture wasn't contrived as in I didn't draw a parabola on the floor but I saw a parabola on this chair. I hope that explained everything. Okay back to our new unit.
Well we're back from spring break and everybody is ready to do some math! Just as a reminder everybody's Flickr pictures are due on the midnight of Friday. Just post a direct link to your Flickr page onto the blog. Remember to tag it with your name, flickr, and trigonometry. For the picture do not contrive it or draw it and say that it is math. It is math but it isn't naturally occuring. The picture must exist as it is naturally. One example of a contrived shot would be a drawing of a parabola on a piece of paper. It is drawn but a natural one would be on that I had taken last year:
My picture wasn't contrived as in I didn't draw a parabola on the floor but I saw a parabola on this chair. I hope that explained everything. Okay back to our new unit. Well we continued on what we did on the last day of class before spring break. We were asked to find other ways to write down several numbers with exponents and only exponents. Mr.K showed us many examples such as using negative exponents. Like in grade 9 math, negative exponents means find the power of the reciprocal of the number. For example:
The number 4 could be written as 16 to the power of 1/2 which is the same as the square root of 16. The second example 1/16 is a fraction so to get from that into a number with a denominator of 1 instead just use a negative exponent to get the reciprocal of 1/16. Then rest of the exponent says 1/2 meaning to get the square root of 16 which is 4. As a reminder of exponents and powers lets look at this picture of the anatomy of a power:
So a is the base which was 16 in the example above, b is the exponent which was 1/2 above and c is the power which is the result, 4.
Okay after find some of those we went back to solving for x but x was in a different place. x was an exponent. If you look at the slides 4 and 5 x is in the exponent spot. So to solve these just try to get the same base on both sides by using the techniques used to find other forms of numbers in exponents like earlier explained. When that is complete since the bases are equal then so must the exponents. So seperately just solve for x like in basic algebra if needed. Sometimes you end up with an algebraic expression like 2x-1 as an exponent. So if the base that has that exponent needs to be changed into another base make sure to multiply the new exponent into 2x-1. Then again solve for x.
Afternoon class. We had a pre-test on identities. Like the normal procedure several minutes to complete the pre-test and then we got into groups and work as a team to solve the questions not completed earlier. After handing in the test we went over the questions. The slides for the pretest is on the second slide of April 7, 2008. The first question nobody had trouble with but we went over the second. The questions asked for the sin(Π+Θ). So first to find sine we first found the sine of Θ which was the y-coordinate, n, over the square root of m squared plus n squared. But the question asked for the sin(Π+Θ) so add that value found earlier to Π and we get a negative value which was answer a. The 3rd question was simple so we did not go over it but the trick to do is was to draw a triangle and find the lengths of the sides. Then you should notice that A and B are the same angle meaning they have the same sides. Then just use the difference identity for cosine and all shall be found well just the answer really. Okay the next question was one that we went over. First just treat this like a normal algebraic equation for a quadratic and factor it. Then we get two answers but one is rejected because it isn't in the domain specified in the question. Then the other one was accepted, which was the answer. Now onto the long answer.
The long answer question was a pretty difficult. The tricky part was actually seeing the double identity on the left side. Change the sin2x into its identity and for the cosine one there are three different identities. On the sixth slide there is only one that would give us a tan by itself and it was the third one on the slide. Everything reduces nice and it leaves with tanx. The right side is a simple identity and we get tan also on the right side. Yay we proved it so remember QED.
We went back to practicing more identity problems. the one we had to do is on the next slide. We had to solve for x. All we have to do is just notice the identity on the left side and change the sin^2 (x) into 1-cos^2(x) and then it becomes a quadratic. Just solve for x like normal. The next slide shows how to find a variable on the calculator. So all we have to do is plug the left side of the equation onto Y1 and the right side onto Y2 and find the intercepts. Make sure to change the window's max and min values into -2Π to 2Π. The intercepts are the answers.
The next few lines are just identity practice for tomorrow's test. So good luck on the test everyone and homework is exercise #20 omit questions #10,11,12. The next scribe will be Eleven. Good night everybody and don't forget the flickr pics.
The number 4 could be written as 16 to the power of 1/2 which is the same as the square root of 16. The second example 1/16 is a fraction so to get from that into a number with a denominator of 1 instead just use a negative exponent to get the reciprocal of 1/16. Then rest of the exponent says 1/2 meaning to get the square root of 16 which is 4. As a reminder of exponents and powers lets look at this picture of the anatomy of a power:
So a is the base which was 16 in the example above, b is the exponent which was 1/2 above and c is the power which is the result, 4.Okay after find some of those we went back to solving for x but x was in a different place. x was an exponent. If you look at the slides 4 and 5 x is in the exponent spot. So to solve these just try to get the same base on both sides by using the techniques used to find other forms of numbers in exponents like earlier explained. When that is complete since the bases are equal then so must the exponents. So seperately just solve for x like in basic algebra if needed. Sometimes you end up with an algebraic expression like 2x-1 as an exponent. So if the base that has that exponent needs to be changed into another base make sure to multiply the new exponent into 2x-1. Then again solve for x.
Afternoon class. We had a pre-test on identities. Like the normal procedure several minutes to complete the pre-test and then we got into groups and work as a team to solve the questions not completed earlier. After handing in the test we went over the questions. The slides for the pretest is on the second slide of April 7, 2008. The first question nobody had trouble with but we went over the second. The questions asked for the sin(Π+Θ). So first to find sine we first found the sine of Θ which was the y-coordinate, n, over the square root of m squared plus n squared. But the question asked for the sin(Π+Θ) so add that value found earlier to Π and we get a negative value which was answer a. The 3rd question was simple so we did not go over it but the trick to do is was to draw a triangle and find the lengths of the sides. Then you should notice that A and B are the same angle meaning they have the same sides. Then just use the difference identity for cosine and all shall be found well just the answer really. Okay the next question was one that we went over. First just treat this like a normal algebraic equation for a quadratic and factor it. Then we get two answers but one is rejected because it isn't in the domain specified in the question. Then the other one was accepted, which was the answer. Now onto the long answer.
The long answer question was a pretty difficult. The tricky part was actually seeing the double identity on the left side. Change the sin2x into its identity and for the cosine one there are three different identities. On the sixth slide there is only one that would give us a tan by itself and it was the third one on the slide. Everything reduces nice and it leaves with tanx. The right side is a simple identity and we get tan also on the right side. Yay we proved it so remember QED.
We went back to practicing more identity problems. the one we had to do is on the next slide. We had to solve for x. All we have to do is just notice the identity on the left side and change the sin^2 (x) into 1-cos^2(x) and then it becomes a quadratic. Just solve for x like normal. The next slide shows how to find a variable on the calculator. So all we have to do is plug the left side of the equation onto Y1 and the right side onto Y2 and find the intercepts. Make sure to change the window's max and min values into -2Π to 2Π. The intercepts are the answers.
The next few lines are just identity practice for tomorrow's test. So good luck on the test everyone and homework is exercise #20 omit questions #10,11,12. The next scribe will be Eleven. Good night everybody and don't forget the flickr pics.
My BOB for Identities
Once again it is time to write a BOB on one of our units. It so happens this time around, the lucky winner is Identities. This unit allowed us to sharpen up our algebra skills(I liked that) as we were "algebraically massaging" expressions.
This was rather an intriguing unit because it made us think "outside the box" and try to be more "elegant" with our solutions. When I say elegant, I don't mean we draw a nice unicorn beside our problems. I mean that we should try to solve the identities with the least number of steps without skipping any.
I found it easier when writing everything in terms of sine and cosine. Although this might not be the best way to come at things because we want to be more elegant.
A big struggle in the unit for me was my own work ethic. Sometimes you think you are ready, but you're not... I feel as though I didn't get enough practice in proving identities. Everything else was a breeze. Well to solve my dilemna there is only one thing to do: practice makes perfect.
That is all.
This was rather an intriguing unit because it made us think "outside the box" and try to be more "elegant" with our solutions. When I say elegant, I don't mean we draw a nice unicorn beside our problems. I mean that we should try to solve the identities with the least number of steps without skipping any.
I found it easier when writing everything in terms of sine and cosine. Although this might not be the best way to come at things because we want to be more elegant.
A big struggle in the unit for me was my own work ethic. Sometimes you think you are ready, but you're not... I feel as though I didn't get enough practice in proving identities. Everything else was a breeze. Well to solve my dilemna there is only one thing to do: practice makes perfect.
That is all.
Friday, April 4, 2008
BOB For Identities
.. I almost forgot that I didn't blog yet. ^^;; Goodness.
So Identities. I was scared of this unit, because I was talking to some people that took this class and they kept telling me that it was confusing and hard and all that jazz, so I freaked out. But it's actually not that bad. I liked this unit, even though it did get confusing in some areas.
Proving identities sometimes took such a long time. I asked myself so many times whether it WAS the same because it took so long. It's like a puzzle, or wait. I remember Justus saying in class, 'this is like a maze', or something like that.
What I need to improve on is applying identities. Sometimes I don't see it, and even though I still get the answer in the end, it doesn't look as 'elegant' as it could be. But I guess that just has to do with more practice. I just need to encourage myself to try and not just give up when I see something that looks complicated. The whole point is to experiment and that's what I need to do.
So uhm, yes. I guess that is it. Test on Monday. Goodluck to all. =)
So Identities. I was scared of this unit, because I was talking to some people that took this class and they kept telling me that it was confusing and hard and all that jazz, so I freaked out. But it's actually not that bad. I liked this unit, even though it did get confusing in some areas.
Proving identities sometimes took such a long time. I asked myself so many times whether it WAS the same because it took so long. It's like a puzzle, or wait. I remember Justus saying in class, 'this is like a maze', or something like that.
What I need to improve on is applying identities. Sometimes I don't see it, and even though I still get the answer in the end, it doesn't look as 'elegant' as it could be. But I guess that just has to do with more practice. I just need to encourage myself to try and not just give up when I see something that looks complicated. The whole point is to experiment and that's what I need to do.
So uhm, yes. I guess that is it. Test on Monday. Goodluck to all. =)
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!
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