SOLUTION: On the unit circle the coordinates of P(theta) are (0.588, 0.809). Find the coordinates of P(2theta). Please help.

Question 393295: On the unit circle the coordinates of P(theta) are (0.588, 0.809). Find the coordinates of P(2theta). Please help.
Answer by jsmallt9(3759) (Show Source): You can put this solution on YOUR website!
(To simplify matter I am going to use "x" instead of "theta". So just replace my x's with theta's.)

Since these points are on the unit circle, the x-coordinates are the cos of the angle to that point and the y-coordinates are the sin of the angle to that point. (This is so because the hypotenuse is always a 1 on the unit circle!) So:
cos(x) = 0.588 and sin(x) = 0.809

To find the coordinates of P(2x) we nned to find the cos(2x) and the sin(2x). Fortunately there are the double angle formulas which express these values in terms of cos(theta) and sin(theta):
cos(2x) = cos²(x) - sin²(x)
and
sin(2x) = 2sin(x)cos(x)
All we have to do is subsitute in the values for cos(x) and sin(x):
cos(2x) = (0.588)² - (0.809)²
and
sin(2x) = 2(0.809)(0.588)
I'll leave it up to you to simplify these expressions. The decimal you get for cos(2x) will be the desired x-coordinate and the decimal you get for sin(2x) will be the desired y-coordinate.
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