Question

In Problems 37-44, find the exact value of each expression.

$cs{c}^{-1}\left(\sqrt{2}\right)$

Short answer

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

Step-by-step solution

Step 1. Write the given information.

The given expression is:$cs{c}^{-1}\left(\sqrt{2}\right)$.

Use the trigonometric properties.

Using the property: $x=cs{c}^{-1}\left(y\right)\Rightarrow cscx=y$.

So,

$$\begin{gathered} \theta =cs{c}^{-1}\left(\sqrt{2}\right) \\ \Rightarrow csc\theta =\sqrt{2} \end{gathered}$$.

Use the trigonometric property.

Using the trigonometric property:

$\sin x=\frac{1}{cscx}\Rightarrow cscx=\frac{1}{\sin x}$.

Thus,

$$\begin{gathered} \sin \left(\theta \right)=\frac{1}{csc\theta } \\ \Rightarrow \sin \left(\theta \right)=\frac{1}{\sqrt{2}} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{4} \end{gathered}$$.