Question

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

$\csc \left[{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$

Short answer

The value of the expression $\csc \left[{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=2$.

Step-by-step solution

Step 1. Given information.  

The given expression is:

$\csc \left[{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$

Let's take $\theta ={\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

$\cos \theta =-\frac{\sqrt{3}}{2}$

$0\le \mathrm{\theta }\le \mathrm{\pi }$is the domain of the function.

Step 2. Find the value of θ.

Initially, we have function ${\cos }^{-1}$and the bounds of cos function are going to be in the interval role="math" localid="1646466354693" $\left[0,\mathrm{\pi }\right]$. It represents the range of ${\cos }^{-1}$.

By using the unit circle, we can say that $\theta$must be equal to $\frac{5\mathrm{\pi }}{6}$, so that we can get $-\frac{\sqrt{3}}{2}$.

$$\begin{gathered} {\cos }^{-1}\theta =-\frac{\sqrt{3}}{2} \\ {\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=\frac{5\mathrm{\pi }}{6} \end{gathered}$$

Step 3. Find the exact value of the expression.

$$\begin{gathered} \csc \left[{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\csc \left(\frac{5\mathrm{\pi }}{6}\right) \\ \csc \left(\mathrm{\theta }\right)=\frac{1}{\sin \mathrm{\theta }} \\ \csc \left(\frac{5\mathrm{\pi }}{6}\right)=\frac{1}{\sin \left(\frac{5\mathrm{\pi }}{6}\right)} \end{gathered}$$

By using the unit circle, we know that $\sin \left(\frac{5\mathrm{\pi }}{6}\right)=\frac{1}{2}$

role="math" localid="1646466896004" $$\begin{gathered} \csc \left(\frac{5\mathrm{\pi }}{6}\right)=\frac{1}{{\displaystyle \frac{1}{2}}} \\ =2 \end{gathered}$$

Therefore,$\csc \left[{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=2$.