CPM Homework Help : PC3 Problem 8-137

If $\cos(x)=-\frac{4}{5}$ and $\sin\left(x\right)<0$ , calculate exact values of $\sin\left(2x\right)$ , $\cos\left(2x\right)$ , $\sin\left(\frac{x}{2}\right)$ , and $\cos\left(\frac{x}{2}\right)$ . First draw a triangle in the unit circle, calculate the values of the missing sides, and then determine if the value of the function you are using in the identities is positive or negative based on the location of the angle.

Hint:

Double-Angle Formulas
$\sin\left(2α\right) = 2\sin\left(α\right)\cos\left(α\right)$
$\cos\left(2α\right) = \cos^{2}\left(α\right) − \sin^{2}\left(α\right)$

Half-Angle Formulas

$\sin\left ( \frac{\theta }{2} \right )=\pm \sqrt{\frac{1-\cos(\theta)}{2}}$

$\cos\left(\frac{\theta }{2} \right )=\pm \sqrt{\frac{1+\cos(\theta)}{2}}$

More Help:

Draw a unit circle and include a right triangle matching the given conditions.
Use the Pythagorean Theorem to determine the value of the missing leg.