Question

In Problems 1– 8, find the exact value of each expression. Do not use a calculator.

$se{c}^{-1}\sqrt{2}$

Short answer

The exact value is $\frac{\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Definition

The inverse secant function $y=se{c}^{-1}x$means that $x=secy$where $\left|x\right|\ge 1$and $0\le y\le \mathrm{\pi }$, $y\ne \frac{\mathrm{\pi }}{2}$.

Step 2. Find the exact value

Let $\theta =se{c}^{-1}\sqrt{2}$, so using the definition of inverse secant function it can be written as $\sqrt{2}=sec\theta$where $0\le \theta \le \mathrm{\pi }$, $\theta \ne \frac{\mathrm{\pi }}{2}$.

In this interval the angle for which the secant value is $\sqrt{2}$is $\frac{\mathrm{\pi }}{4}$.

So the value of $se{c}^{-1}\sqrt{2}$is $\frac{\mathrm{\pi }}{4}$.