Question

Find the domain of inverse trigonometric functions?

Answer 1

Here in this question, we need to find the domain of the inverse trigonometric functions. The domain refers to the set of possible values of x for which the function will be defined and the range refers to the possible range of values that the function can attain for those values of x which are in the domain of the function.

Complete step by step solution:
The various trigonometric functions are ${\sin ^{ - 1}}$, ${\cos ^{ - 1}}$, ${\tan ^{ - 1}}$, ${\csc ^{ - 1}}$, ${\sec ^{ - 1}}$ and ${\cot ^{ - 1}}$.
We do know that a function that has an inverse has exactly one output for exactly one input. The domain of an inverse trigonometric function is equal to the range of its counter trigonometric function.
The domain of inverse trigonometric functions, ${\sin ^{ - 1}}$,${\cos ^{ - 1}}$ and ${\tan ^{ - 1}}$ is equal to the range of the corresponding trigonometric functions,$\sin \theta$,$\cos \theta$ and $\tan \theta$.
Therefore,
Domain of ${\sin ^{ - 1}}$ and ${\cos ^{ - 1}}$ is \left\\{ {x \in \mathbb{R}: - 1 \leqslant x \leqslant 1} \right\\} since$- 1 \leqslant \sin \theta \leqslant 1$ and$- 1 \leqslant \cos \theta \leqslant 1$for all $\theta \in \mathbb{R}$.
Domain of ${\tan ^{ - 1}}$ is $\mathbb{R}$since the range of $\tan \theta$ is the whole of $\mathbb{R}$.
Domain of ${\csc ^{ - 1}}$ and ${\sec ^{ - 1}}$ is \left\\{ {x \in \mathbb{R}:x \leqslant - 1\;or\;1 \leqslant x} \right\\}.
Domain of ${\cot ^{ - 1}}$ is $\mathbb{R}$.

Note: An independent set of those values for a given function which on substitution always gives real value of result but set of value of range is depending upon the value of the set of domains as it is necessary that all ranges must have domain is known as domain. When a function is defined, its domain and co-domain are also mentioned. The domain and co-domain need not be different. They may be the same as well.