Question

use a reference triangle to find the given angle.
cos^{-1}left(\frac{-1}{2}
ight)
cos^{-1}left(\frac{-1}{2}
ight)=square
(type an exact answer in terms of (pi).)

Explanation:

Step1: Recall the range of inverse - cosine

The range of $y = \cos^{-1}(x)$ is $[0,\pi]$.

Step2: Find the angle in the unit - circle

We know that $\cos\theta=-\frac{1}{2}$. In the unit - circle, in the range $[0,\pi]$, when $\cos\theta =-\frac{1}{2}$, the reference triangle in the second quadrant has an adjacent side to the angle $\theta$ of length $- 1$ (in the $x$ - direction) and a hypotenuse of length $2$. The angle $\theta$ that satisfies $\cos\theta=-\frac{1}{2}$ and is in the range $[0,\pi]$ is $\frac{2\pi}{3}$.

Answer:

$\frac{2\pi}{3}$