Question

Solve $2{\sin ^{ - 1}}(\dfrac{3}{5}) + {\cos ^{ - 1}}(\dfrac{{24}}{{25}})$
A. $\dfrac{\pi }{2}$
B. $\dfrac{{2\pi }}{3}$
C. $\dfrac{{5\pi }}{3}$
D. None of these

Answer 1

Hint: It is necessary to remember a few fundamental formulas to solve trigonometric equations. To answer this particular question, we need to remember the formulas of inverse trigonometric ratios in addition to the usual trigonometric formulas.

Formula Used:
${\sin ^2}x + {\cos ^2}x = 1$
$\cos 2A = 2{\cos ^2}A - 1$

Complete step by step Solution:
Let us suppose that $2{\sin ^{ - 1}}(\dfrac{3}{5}) = A$
Then we have, ${\sin ^{ - 1}}(\dfrac{3}{5}) = \dfrac{A}{2}$
$ \Rightarrow \sin \dfrac{A}{2} = \dfrac{3}{5}$
Since ${\sin ^2}x + {\cos ^2}x = 1$
$ \Rightarrow {\cos ^2}x = 1 - {\sin ^2}x$
$ \Rightarrow {\cos ^2}\left( {\dfrac{A}{2}} \right) = 1 - {\sin ^2}\left( {\dfrac{A}{2}} \right)$
                         $ = 1 - {\left( {\dfrac{3}{5}} \right)^2}$ $ = 1 - \dfrac{9}{{25}}$
$ \Rightarrow \cos \dfrac{A}{2} = \sqrt {(1 - \dfrac{9}{{25}})} $
$ \Rightarrow \cos \dfrac{A}{2} = \sqrt {\dfrac{{16}}{{25}}} = \dfrac{4}{5}$
Now we will find the value of $\cos A$
We know that, $\cos 2A = 2{\cos ^2}A - 1$
$ \Rightarrow \cos A = 2{\cos ^2}(\dfrac{A}{2}) - 1$
Therefore, we have, $\cos A = 2(\dfrac{{16}}{{25}}) - 1$
$ \Rightarrow \cos A = \dfrac{7}{{25}}$
Thus, $A = {\cos ^{ - 1}}(\dfrac{7}{{25}})$
Now,
$2{\sin ^{ - 1}}(\dfrac{3}{5}) + {\cos ^{ - 1}}(\dfrac{{24}}{{25}}) = {\cos ^{ - 1}}(\dfrac{7}{{25}}) + {\cos ^{ - 1}}(\dfrac{{24}}{{25}})$
Using the formula ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y = {\cos ^{ - 1}}[xy - \sqrt {(1 - {x^2})} \sqrt {(1 - {y^2})} ]$
${\cos ^{ - 1}}(\dfrac{7}{{25}}) + {\cos ^{ - 1}}(\dfrac{{24}}{{25}}) = {\cos ^{ - 1}}[(\dfrac{7}{{25}})(\dfrac{{24}}{{25}}) - \sqrt {1 - \dfrac{{49}}{{625}}} \sqrt {1 - \dfrac{{576}}{{625}}} ]$
$ = {\cos ^{ - 1}}[\dfrac{{168}}{{625}} - \sqrt {\dfrac{{576}}{{625}}} \sqrt {\dfrac{{49}}{{625}}} ]$
On further solving we get,
${\cos ^{ - 1}}(\dfrac{7}{{25}}) + {\cos ^{ - 1}}(\dfrac{{24}}{{25}}) = {\cos ^{ - 1}}[\dfrac{{168}}{{625}} - (\dfrac{{24}}{{25}})(\dfrac{7}{{25}})]$
$ = {\cos ^{ - 1}}[\dfrac{{168}}{{625}} - \dfrac{{168}}{{625}}]$
$ = {\cos ^{ - 1}}0$
$ = {\cos ^{ - 1}}(\cos \dfrac{\pi }{2})$
$ = \dfrac{\pi }{2}$

Hence, the correct option is (A).

Note: Assume the value ${\sin ^{ - 1}}\dfrac{3}{5}$ with some suitable constant for easy calculation. After solving the question, in the end, we need to replace the variable we have taken with the inverse of sin which will give us the final answer.