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.
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