Query 2

Query 2

This is posted late due to an error. Should have been posted on 01-31. I believe I emailed you a response. Sorry for the confusion.

Good work. See my comments on the first problem and let me know if you have questions.

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Precalculus II Asst # 2 01-30-2005

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15:27:08

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**** query ch. 5.2 # 102 f(x) = cos(x), f(a) =1/4, find f(-a), f(a) + f(a+2`pi) + f(a - 2 `pi)

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15:28:48 f(-a) = -1/4 f(a) =1/4, find f(-a), f(a) + f(a+2`pi) + f(a - 2 `pi)=3/4

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15:28:50

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**** What is the value of f(a) + f(a+2`pi) + f(a - 2 `pi) and how did you use the even-odd and/or periodic properties of the function to obtain your result?

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15:31:12 Well Sir, I couldn't relate the first part where you stated f(x)=cos x and f(a) =1/4 f(x) cos x is even but that where I stop.

The cosine is an even function, meaning that f(-x) = f(x) for every x. That is related to the fact that if you move around the circle clockwise through a given angle, the x coordinate is the same as if you had moved counterclockwise.

So in any case f(-a) = f(a).

Furthermore if you move from angular position a to angular position a + 2 pi, you will have made a complete trip around the circle and you will be right where you started. So cos(a + 2 pi) is the same as cos(a) and f(a + 2 pi) = f(a).

Similarly f(a - 2 pi) = f(a).

Thus f(a) + f(a+2`pi) + f(a - 2 `pi) = 3 * f(a).

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15:31:13

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**** query (no summary needed)

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15:31:29 ok

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15:31:31

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**** How does the circular model demonstrate the periodic nature of the trigonometric functions? Be specific.

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15:33:56 The unit circe when broke into a series of 8 or 12 sectors yields the same coordinates when any one value (coordinated) has 2Pi added to it...

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15:33:56

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**** How does the circular model demonstrate the even or odd nature of the sine and cosine functions? Be specific.

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15:41:33 We are told that an even function is symmetriacl about the y axis and odd function symmetric about the origin..The cosine and secant (its reciprical, that is to say 1/cosine) are the only even functions of the unit circle.. When graphed you will see the sine function go through the origin where the cosine will not..The secant will have vertical asymptote...

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15:41:34

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**** Can you very quickly sketch on a reference circle the angles which are multiples of `pi/6 and immediately list the sine and cosine of each? Can you do the same for multiples of `pi/4? (It's OK to answer honestly but be prepared to have to do this on a test, and remember that this task is central to understanding the trigonometric functions; if you've reached this point without that skill you have already wasted a lot of time by not knowing something you need to know to do what you're trying to do).

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15:42:58 Yes, very much so.. Not much in it really !!

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15:43:02

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**** Query Add comments on any surprises or insights you experienced as a result of this assignment.

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15:48:53 This is wonderful knowledge to have aquired. I have started to use the principles of the Trg. functions in conjunctions with the definitions of Sine, Cos and Tan to solve for different problems etc..

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15:49:01 Your response has been entered.

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15:49:09 Your response has been entered. ok

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