Question

The principal value of ${\cos }^{-1}\left\{\left(\frac{1}{\sqrt{2}}\right)\left[\cos \left(\frac{9\mathrm{\pi }}{10}\right)-\sin \left(\frac{9\mathrm{\pi }}{10}\right)\right]\right\}$is

• A. $\frac{3\mathrm{\pi }}{10}$
• B. $\frac{17\mathrm{\pi }}{20}$
• C. $\frac{7\mathrm{\pi }}{10}$
• D. None of these

Answer 1

The correct option is B

$\frac{17\mathrm{\pi }}{20}$

Explanation for the correct option

${\cos }^{-1}\left\{\left(\frac{1}{\sqrt{2}}\right)\left[\cos \left(\frac{9\mathrm{\pi }}{10}\right)-\sin \left(\frac{9\mathrm{\pi }}{10}\right)\right]\right\}$

$={\cos }^{-1}\left\{\left(\frac{1}{\sqrt{2}}\right)\left[\cos \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{10}\right)-\sin \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{10}\right)\right]\right\}$ { Domain of ${\cos }^{-1}x$ is $[-1,1]$ }

$$\begin{gathered} ={\cos }^{-1}\left\{\left(\frac{1}{\sqrt{2}}\right)\left[-\cos \left(\frac{\mathrm{\pi }}{10}\right)-\sin \left(\frac{\mathrm{\pi }}{10}\right)\right]\right\} \\ ={\cos }^{-1}\left\{(-1)\left[\frac{1}{\sqrt{2}}\cos \left(\frac{\mathrm{\pi }}{10}\right)+\frac{1}{\sqrt{2}}\sin \left(\frac{\mathrm{\pi }}{10}\right)\right]\right\} \\ ={\cos }^{-1}\left\{(-1)\left[\cos \frac{\mathrm{\pi }}{4}\cos \left(\frac{\mathrm{\pi }}{10}\right)+\sin \frac{\mathrm{\pi }}{4}\sin \left(\frac{\mathrm{\pi }}{10}\right)\right]\right\} \\ ={\cos }^{-1}\left\{(-1)\left[\cos \left(\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{10}\right)\right]\right\}\mathbf{\left[}\mathbf{\because }\mathbf{cosAsinB}\mathbf{+}\mathbf{sinAcosB}\mathbf{=}\mathbf{cos}\mathbf{(}\mathbf{A}\mathbf{-}\mathbf{B}\mathbf{)}\mathbf{\right]} \\ =\mathrm{\pi }-{\cos }^{-1}\left(\cos \frac{3\mathrm{\pi }}{20}\right)=\mathrm{\pi }-\frac{3\mathrm{\pi }}{20}=\frac{17\mathrm{\pi }}{20}\mathbf{[}\mathbf{\because }{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{\left[}\mathbf{cos}\mathbf{(}\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{\right]}\mathbf{=}\mathbf{\pi }\mathbf{-}\mathbf{x}\mathbf{]} \end{gathered}$$

Hence the correct option is option(B).