# How do you evaluate sec^-1(-2) without a calculator?

Sep 30, 2016

$x = \setminus \pm \frac{2 \setminus \pi}{3}$.

#### Explanation:

What do you mean exactly with that exponent $- 1$?

If you mean the power, we have

$\sec \left(x\right) = \frac{1}{\cos} \left(x\right) \setminus \implies {\sec}^{- 1} \left(x\right) = \frac{1}{\frac{1}{\cos} \left(x\right)} = \cos \left(x\right)$

Thus, ${\sec}^{- 1} \left(- 2\right) = \cos \left(- 2\right)$, and I see no way to calculate it manually.

If you mean the inverse function, we want to find a number $x$ such that $\sec \left(x\right) = - 2$, or if you prefer, $\frac{1}{\cos} \left(x\right) = - 2$

This clearly leads to $\cos \left(x\right) = - \frac{1}{2}$, and it is a known angle: $x = \setminus \pm \frac{2 \setminus \pi}{3}$, depending on which interval you choose to invert the cosine function.