Question

How do you evaluate ${\sin ^{ - 1}}\left( 2 \right)$ ?

Answer 1

Hint : In order to find the value of an inverse trigonometric function, we first set up the principal value branch in which the value of the inverse trigonometric function should lie. The principal value branch is basically the chosen range of the inverse function.
For ${\sin ^{ - 1}}$ function, the principal value branch is $\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$ .
For ${\cos ^{ - 1}}$ function, the principal value branch is $\left[ {0,\pi } \right]$ .
For ${\tan ^{ - 1}}$ function, the principal value branch is $\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$ .

Complete step-by-step answer :
According to definition of inverse ratio,
If the value of x is ${\sin ^{ - 1}}\left( 2 \right)$ ,
Then, $\sin x = 2$ is the trigonometric equation that corresponds to the value of x assumed where the value of x lies in the range $\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$ as the principal value branch of ${\sin ^{ - 1}}$ function is $\left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$ .
However, we know that the value of the function $\sin x$ lies between $- 1$ and $1$ as the range of the function $\sin x$ is $\left[ { - 1,1} \right]$ .
So, $\sin x = 2$ is not possible and has no real solution for values of x.
Hence, the value of ${\sin ^{ - 1}}\left( 2 \right)$ is undefined or there is no real value of function ${\sin ^{ - 1}}\left( 2 \right)$ .

Note : The basic inverse trigonometric functions are used to find the missing angles in right triangles. While the regular trigonometric functions are used to determine the missing sides of the right-angled triangles. Besides the trigonometric functions and inverse trigonometric functions, we also have some rules related to trigonometry such as the sine rule and cosine rule.