Question

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

${\sin }^{-1}1$

Short answer

The exact value is $\frac{\mathrm{\pi }}{2}$.

Step-by-step solution

Step 1. Definition

The inverse sine function $y={\sin }^{-1}x$means that $x=\sin y$where $-1\le x\le 1$and $-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$.

Step 2. Find the exact value

Let $\theta ={\sin }^{-1}1$, so using the definition of inverse sine function it can be written as $1=\sin \theta$where $-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$.

In this interval the angle for which the sine value is $1$is $\frac{\mathrm{\pi }}{2}$.

So the value of${\sin }^{-1}1$is$\frac{\mathrm{\pi }}{2}$.