Question

If ${\sin }^{-1}\left(\frac{x}{13}\right)+{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi }{2}$ then the value if $x$ is

• A. $5$
• B. $4$
• C. $10$
• D. $11$

Answer 1

The correct option is A

$5$

Explanation for the correct option.

Given that, ${\sin }^{-1}\left(\frac{x}{13}\right)+{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi }{2}$.

$\begin{array}{rcl}cose{c}^{-1}\left(\frac{13}{12}\right)& =& \frac{\pi }{2}-{\sin }^{-1}\left(\frac{x}{13}\right)\\ cose{c}^{-1}\left(\frac{13}{12}\right)& =& {\cos }^{-1}\left(\frac{x}{13}\right)[{\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}]\end{array}$

Now convert $cose{c}^{-1}\left(\frac{13}{12}\right)$ in terms of ${\cos }^{-1}$.

As we know that, $\text{cosec}\theta \text{=}\frac{13}{12}=\frac{\text{Hypotenuse}}{\text{Perpendicular}}$

$\begin{array}{c}\text{Base}=\sqrt{{\text{Hypotenuse}}^2-{\text{Perpendicular}}^2}\\ =\sqrt{169-144}\\ =\sqrt{25}\\ =5\end{array}$

$\begin{array}{c}\therefore \cos \left(\theta \right)=\frac{\text{Base}}{\text{Hypotenuse}}\\ =\frac{5}{13}\\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{5}{13}\right)\end{array}$

$\begin{array}{rcl}\therefore {\cos }^{-1}\left(\frac{5}{13}\right)& =& {\cos }^{-1}\left(\frac{x}{13}\right)\\ \frac{5}{13}& =& \frac{x}{13}\mathrm{comparing}\\ \mathrm{x}& =& 5\end{array}$
Hence, option A is correct.