#$&* course MTH 164 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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00:05:16 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If f(a)=1/4 then ¼+1/4+1/4=3/4 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** the cosine is an even function, with f(-a) = f(a) so if f(a) = 1/4, f(-a) = 1/4. The idea of periodicity is that f(a+2`pi) = f(a), and the same for f(a-2`pi). Since f(a) = 1/4, all these terms are 1/4 and f(a) + f(a+2`pi) + f(a - 2 `pi) = 1/4 + 1/4 + 1/4 = 3/4. It is helpful to visualize the situation on a unit circle. If a is the angular position on the unit circle, then at angular position -a the x coordinate will be the same. So the cosine of a and of -a is the same. Angular positions a + 2 `pi and a - 2 `pi put you at the same location on the circle, since 2 pi corresponds to one complete revolution. So any trigonometric function will be the same at a, a + 2 pi and a - 2 pi. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: query (no summary needed) **** How does the circular model demonstrate the periodic nature of the trigonometric functions? Be specific.
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00:13:53 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If you go the whole way around the circle you would be at the same point, so you would have the same functions even if you’re going around multiple times. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ********************************************* Question:** the circular model demonstrates the periodic nature of the trigonometric functions because if you go all the way around the circle you end up at the same point, giving you the same values of the trigonometric functions, even though in going around an additional time the angle has changed by 2 `pi. • This is the case no matter how many times you go around. • Every time the angle changes by 2 `pi you find yourself at the same point with the same values. **
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00:13:54
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: Question: **** How does the circular model demonstrate the even or odd nature of the sine and cosine functions? Be specific.
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00:16:35 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: With cosine if you go around the circle through the angular distance, you will end up at the same x coordinate. With sine, you will have the odd values of it. \ confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Question:** The answer can be pictured in terms of 2 ants, one going counterclockwise and the other clockwise. • The cosine is the x coordinate of the reference point. Since we start at the positive x axis, it doesn't matter whether we go clockwise or counterclockwise through the given angular distance, we end up with the same x coordinate. • The sine function being the y coordinate, clockwise motion takes us first to negative values of the sine while counterclockwise motion takes us first to positive values of the sine. Thus the sine is odd. ** STUDENT COMMENT: Ok, by looking at a complete unit circle I can see what you are talking about. Sometimes I have trouble visualizing this. INSTRUCTOR RESPONSE: Sketch it every time you need it; you will probably soon begin to visualizing it without the need of the sketch, but if not it's always fine to revert to sketching.<
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00:16:36
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: **** 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. You should 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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00:17:51 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Yes. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Multiples of pi/6 give you magnitudes 0, 1/2, sqrt(3)/2 = .87 approx., and 1. It is clear from a decent sketch which gives you which, and when the result is positive and when negative. Multiples of pi/4 given you magnitudes 0, sqrt(2)/2 = .71 and 1 approx., and again a good sketch makes it clear which is which. ** STUDENT QUESTION: I am quickly realizing how important this is. While I can slowly work my way through the calculations, I think I might be better off memorizing the coordinates. Is that the goal? Should I try to memorize my pi coordinates and the sine and cosine of each? INSTRUCTOR RESPONSE: If you work them out every time you need them, you'll soon be able to work them out quickly, and in the process you will not only memorize them but will understand how to quickly confirm them out any time your memory fails. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: **** Query Add comments on any surprises or insights you experienced as a result of this assignment.
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00:23:45 i was suprised by the fact that this assignment was shorter than the other yet somehow it seemed more complex.
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Miscellaneous comments, questions, etc.: 117. If f(theta) = sec(theta) and f(a) = -4, find the exact value of A) f(-a) = f(-(-4)) = 4 b) f(a) + f(a+2pi) +(a+4pi) = -8.8584 ****( I checked the answer in the back of the book and I know it is -12. I read in the book that you can ignore 2pi and multiples of 2pi because they represent complete revolutions, but I’m not sure I understand why.