Question

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

Answer 1

Here, we have to evaluate the inverse of the sine function. We will use the trigonometric ratio and by rewriting the equation, we will evaluate the inverse of the sine function. A trigonometric equation is defined as an equation involving the trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables.

Formula Used:
Trigonometric ratio: $\sin 90^\circ = 1$.

Complete Step by Step Solution:
We are given a trigonometric equation ${\sin ^{ - 1}}\left( 1 \right)$.
We know that a trigonometric ratio of $\sin 90^\circ$ is $1$.
$\sin 90^\circ = 1$
Taking sine inverse on both sides, we get
$\Rightarrow 90^\circ = {\sin ^{ - 1}}\left( 1 \right)$
$\Rightarrow {\sin ^{ - 1}}\left( 1 \right) = 90^\circ$
The inverse of sine $1$ is $90^\circ$.

Therefore, the inverse of sine $1$ is $90^\circ$ or $\dfrac{\pi }{2}$.

Additional Information:
We know that we have many trigonometric identities that are related to all the other trigonometric equations. We need to remember that the trigonometric ratio and the co-trigonometric ratio is always reciprocal to each other. A trigonometric ratio is used to find the relationships between the sides of a right-angle triangle and also helps in finding the lengths of the triangle.

Note: The inverse trigonometric function is used to find the missing angles in a right-angled triangle whereas the trigonometric function is used to find the missing sides in a right-angled triangle. The range of the arcsine of the angle lies between $- \dfrac{\pi }{2}$ radians and $\dfrac{\pi }{2}$ radians. The maximum of the sine of the angle is 1 and at $\dfrac{\pi }{2}$ radians. The basic angles used in solving trigonometric problems are in degrees. The trigonometric angles can also be denoted in Radians