SOLUTION: If 0 < x < pi/2 and tanx = 15/8, find cos2x.

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Question 230621: If 0 < x < pi/2 and tanx = 15/8, find cos2x.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since 0+%3C+x+%3C+pi%2F2, x is in the first quadrant and sin cos and tan are all positive. (2x may end up in the second quadrant but we'll deal with that when the time comes.)

Since the tan function represents the ratio of opposite over adjacent we can use 15 for the opposite side and 8 for the adjacent side.

Next we'll use the Pythagorean Theorem to find the hypotenuse:
15%5E2+%2B+8%5E2+=+%28hyp%29%5E2
225+%2B+64+=+%28hyp%29%5E2
289+=+%28hyp%29%5E2
17 = hypotenuse

Now we want to find the cos(2x). We can use any of the identities for cos(2x):
cos%282x%29+=+%28cos%28x%29%29%5E2+-+%28sin%28x%29%29%5E2
cos%282x%29+=+2%28cos%28x%29%29%5E2+-+1
cos%282x%29+=+1+-+2%28sin%28x%29%29%5E2
The formulas will handle whether or not x ends up in the second quadrant and assign the correct sign. I'll use the first.
cos(x) = adjacent/hypotenuse = 8/17
sin(x) = opposite/hypotenuse = 15/17
cos%282x%29+=+%288%2F17%29%5E2+-+%2815%2F17%29%5E2
cos%282x%29+=+64%2F289+-+225%2F289
cos%282x%29+=+-181%2F289
Since cos(2x) is negative, 2x apparently does end up in the second quadrant.