Question

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

${\cos }^{-1}\left(\sin \frac{5\mathrm{\pi }}{4}\right)$

Short answer

The value of the expression${\cos }^{-1}\left(\sin \frac{5\mathrm{\pi }}{4}\right)=\frac{3\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Write the given information.  

The given expression is:

${\cos }^{-1}\left(\sin \frac{5\mathrm{\pi }}{4}\right)$

Use the trigonometric identities. 

Use the identity $\sin \left(\frac{\mathrm{\pi }}{2}+x\right)=\cos x$,

$$\begin{gathered} {\cos }^{-1}\left(\sin \frac{5\mathrm{\pi }}{4}\right)={\cos }^{-1}\left[\sin \left(\frac{\mathrm{\pi }}{2}+\frac{3\mathrm{\pi }}{4}\right)\right] \\ ={\cos }^{-1}\left[\cos \left(\frac{3\mathrm{\pi }}{4}\right)\right] \end{gathered}$$

Use the identity cos-1cos x=x.

${\cos }^{-1}\left[\cos \left(\frac{3\mathrm{\pi }}{4}\right)\right]=\frac{3\mathrm{\pi }}{4}$

Therefore,${\cos }^{-1}\left(\sin \frac{5\mathrm{\pi }}{4}\right)=\frac{3\mathrm{\pi }}{4}$.