Question

Write the value of cos⁻¹ $\left(\cos \frac{5\mathrm{\pi }}{4}\right)$.

Answer 1

${\cos }^{-1}\left(\cos \frac{5\mathrm{\pi }}{4}\right)\ne \frac{5\mathrm{\pi }}{4}$ as $\frac{5\mathrm{\pi }}{4}$ does not lie between $0\mathrm{and}\mathrm{\pi }$.
We have

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