# How do you evaluate sec^-1(2)?

Aug 13, 2016

${\sec}^{- 1} \left(2\right) = \frac{\pi}{3}$

#### Explanation:

Let the answer be $x$.

Since ${\sec}^{- 1} \left(2\right) = x$, $\sec \left(x\right) = 2$.

As secant is the reciprocal of the cosine function, i.e. $\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$, the following must be true as well.

$\frac{1}{\cos} \left(x\right) = 2$

Or $\cos \left(x\right) = \frac{1}{2}$.

The range of ${\sec}^{- 1} \left(x\right)$ is $\left[0 , \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2} , \pi\right]$.
The range of ${\cos}^{- 1} \left(x\right)$ is $\left[0 , \pi\right]$.

Since the ranges are similar, we can find $x$ using

$x = {\cos}^{- 1} \left(\frac{1}{2}\right) = \frac{\pi}{3}$