Question

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

$\csc \left[{\tan }^{-1}\left(-2\right)\right]$

Short answer

The value of the expression$\csc \left[{\tan }^{-1}\left(-2\right)\right]=-\frac{\sqrt{5}}{2}$.

Step-by-step solution

Step 1. Write the given information.

The given expression is:

$\csc \left[{\tan }^{-1}\left(-2\right)\right]$

Step 2. Find the hypotenuse side.

Let's assume $\theta ={\tan }^{-1}\left(-2\right)$

$\tan \mathrm{\theta }=-2;\tan \mathrm{\theta }=\frac{\mathrm{opposite}\mathrm{side}}{\mathrm{adjacent}\mathrm{side}}$

Now consider a right angle triangle that has its opposite side equal to $-2$and its adjacent side equals to 1 and $\theta$be the angle which is formed by the intersection of the hypotenuse and adjacent sides.

By using Pythagoras theorem, We will get the hypotenuse side as $\sqrt{5}$.

Step 3. Find the value of csc θ.

$$\begin{gathered} \csc \theta =\frac{1}{\sin \theta } \\ \sin \theta =\frac{\mathrm{adjacent}\mathrm{side}}{\mathrm{hypotenuse}\mathrm{side}};\sin \theta =-\frac{2}{\sqrt{5}}, \\ \csc \theta =\frac{1}{\sin \theta } \\ =\frac{1}{-\frac{2}{\sqrt{5}}} \\ =-\frac{\sqrt{5}}{2} \\ \csc \theta =\csc \left[{\tan }^{-1}\left(-2\right)\right] \end{gathered}$$

Therefore, $\csc \left[{\tan }^{-1}\left(-2\right)\right]=-\frac{\sqrt{5}}{2}$.