Question

In Problems 9–36, find the exact value of each expression.

$\sin \left[{\tan }^{-1}\left(-1\right)\right]$

Short answer

The value of the expression$\sin \left[{\tan }^{-1}\left(-1\right)\right]=\left(-\frac{\sqrt{2}}{2}\right)$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$\sin \left[{\tan }^{-1}\left(-1\right)\right]$

Let's take $\theta ={\tan }^{-1}\left(-1\right)$,

$\tan \theta =-1$

$-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$is the domain of the function.

Step 2. Find the value of θ.

Initially, we have function ${\tan }^{-1}$and the bounds of tan function are going to be in the interval $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$. It represents the range of ${\tan }^{-1}$.

By using the unit circle, we can say that $\theta$must be equal to $\left(-\frac{\mathrm{\pi }}{4}\right)$, so that we can get $-1$.

$$\begin{gathered} \tan \theta =-1 \\ {\tan }^{-1}(-1)=\left(-\frac{\mathrm{\pi }}{4}\right) \end{gathered}$$

Step 3. Find the exact value of the expression. 

$\sin \left[{\tan }^{-1}\left(-1\right)\right]=\sin \left(-\frac{\mathrm{\pi }}{4}\right)$

By using the unit circle, we know that $\sin \left(-\frac{\mathrm{\pi }}{4}\right)=\left(-\frac{\sqrt{2}}{2}\right)$

Therefore,$\sin \left[{\tan }^{-1}\left(-1\right)\right]=\left(-\frac{\sqrt{2}}{2}\right)$