CPM Homework Help : PC Problem 6-109

If $\cos x = -\frac{4}{5}$ , $x ∈ [π, \frac { 3 \pi } { 2 }]$ , find exact values for $\sin 2x$ , $\cos 2x$ , $\sin\left(\frac{x}{2}\right)$ , and $\cos\left(\frac{x}{2}\right)$ . Start by drawing a triangle that where $\cos x = \frac{4}{5}$ , find the missing sides and then determine if the value of the function you are using in the formulas 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.