Question

Example 3. Evaluate the following
(i) sin-1 (sin 10)​

Answer 1

Answer:

math]sin^{-1}(sin x)[/math] needs to be understood correctly. Sine being a periodic function, there will be many values [math]y[/math] satisfying [math]sin y= sin x.[/math] When the notation of inverse trigonometric function is used, then its range is restricted. The principal branch of [math]sin^{-1}[/math] is defined as the function [math]sin^{-1}: [-1,1]---> [-pi/2, pi/2].[/math] Therefore, sin^-1(sin x)= x holds for only those x that belong to the interval [math][-pi/2, pi/2].[/math] Since 10 lies between 3pi and 7pi/2, sin^-1(sin 10) will be equal to 10–4pi. It is easy to check that sin(10–4pi)=sin 10, and 10–4pi lies in the interval [math][-pi/2, pi/2].[/math] If you want to know the value of [math]sin^{-1}(sin x),[/math] then draw the graph of sin x, draw the line parallel to x-axis, through the poing (x, sin x), and see where it hits the graph of sin x over the interval [math][-pi/2, pi/2], and its x-coordinate is [/math]the answer you are looking for.