Question

Solve ${\sin ^{ - 1}}\left( {\cos {\text{ x}}} \right)$.

Answer 1

Hint- Here we will proceed by using the one of the property of inverse trigonometric function i.e. $\left( {\cos {\text{ }}\theta } \right){\text{ = sin}}\left( {\dfrac{\pi }{2} - \theta } \right){\text{ }}$. Then we will multiply it with ${\sin ^{ - 1}}$ to get the required result.

Complete step-by-step answer:

As we know that,
 $ \Rightarrow \left( {\cos {\text{ }}\theta } \right){\text{ = sin}}\left( {\dfrac{\pi }{2} - \theta } \right){\text{ }}$
Therefore,
$ \Rightarrow {\sin ^{ - 1}}\left( {\cos {\text{ x}}} \right)$
$ \Rightarrow {\sin ^{ - 1}}\left( {\sin \left( {\dfrac{\pi }{2} + \theta } \right)} \right)$
Also we know that,
${f^{ - 1}}\left( {f\left( x \right)} \right) = x$
Which implies that-
$ = \left( {\dfrac{\pi }{2} + \theta } \right)$
Hence the answer is $\left( {\dfrac{\pi }{2} + \theta } \right)$

Note- In order to solve this type of questions, we must know all the inverse trigonometric functions which are ${\sin ^{ - 1}}\theta ,{\cos ^{ - 1}}\theta ,{\tan ^{ - 1}}\theta ,\cos e{c^{ - 1}}\theta ,{\sec ^{ - 1}}\theta ,{\cot ^{ - 1}}\theta $ as here also we used one of its function i.e. $\left( {\cos {\text{ }}\theta } \right){\text{ = sin}}\left( {\dfrac{\pi }{2} - \theta } \right){\text{ }}$so that we can also tackle similar type of questions.