SOLUTION: find the exact value of cos(2θ)and sin(2θ) if tan(θ) = (-5/12) and 270° < θ < 360°

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Question 1155044: find the exact value of cos(2θ)and sin(2θ) if tan(θ) = (-5/12) and 270° < θ < 360°
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
We know that  270° < θ < 360° means that θ is in quadrant IV.
We also know that the tangent is the opposite/adjacent or y/x.
So we can draw a right triangle in the 4th quadrant so that
the opposite side is the numerator +5 of the tangent (-5/12) 
taken as (+5)/(-12), and the denominator is -12 of the tangent 
taken as (+5)/(-12).  So we have the drawing: 



Next we use the Pythagorean theorem to calculate r:



Fill in the value of r=13:



Now we use the formula for sin(2θ):

sin%282theta%29=2sin%28theta%29cos%28theta%29

and the fact that the sine is opposite/hypotenuse or y/r = (-12)/(13) = -12/13
and the fact that the cosine is adjacent/hypotenuse or x/r = (+5)/(13) = 5/13

sin%282theta%29=2%28-12%2F13%29%285%2F13%29

sin%282theta%29=2%28-12%2F13%29%285%2F13%29=-120%2F169

We use the formula for cos(2θ):

cos%282theta%29=cos%5E2%28theta%29-sin%5E2%28theta%29


cos%282theta%29=%28-5%2F13%292-%28-12%2F13%29%5E2

cos%282theta%29=25%2F169-144%2F169
 
cos%282theta%29=-119%2F169

Edwin