Find sin x 2 , cos x 2 , and tan x 2 from the given information. cos(x) = − 15/ 17 , 180° < x < 270°

Question

Find sin x/ 2 , cos x/ 2 , and tan x /2 from the given information. cos(x) = −  15/ 17 ,   180° < x < 270°

sin(x/2=

cos(x/2)=

tan(x/2)=

Please include steps. Thank you!

Answer 1

We must use the half angle theorems here.

https://www.onlinemathlearning.com/image-files/double-angle-formulas.png

Theta is used to denote an angle here, so just replace that with x. -15/ 17

sin(x/2) = ±√ ((1 - cos(x)) / 2)

cos(x/2) = ±√ ((1 + cos(x)) / 2)

tan(x/2) = ±√ ((1 - cos(x)) / (1 + cos(x))

These formulae have a plus or a minus, but we can't leave that plus minus in the answer. We know 180° < x < 270°. That means 90° < (x / 2) < 270°. This tells us for sure that it falls in Quadrant II. From the unit circle, we know that indicates a positive sine, a negative cosine, and since the sin and cos are opposite signs, tangent is negative.

sin(x/2) = √ ((1 + 15/ 17) / 2)

cos(x/2) = - √ ((1 - 15/ 17) / 2)

tan(x/2) = - √ ((1 + 15/ 17) / (1 - 15/ 17) )