Explain why sin1 Squareroot 1 cos2 theta theta for 0 lesst

Explain why sin^-1 (Squareroot 1 - cos^2 (theta)) = theta for 0 lessthanorequalto theta lessthanorequalto pi/2 by looking at the following triangle What happens when -pi/2 lessthanorequalto theta lessthanorequalto 0?

Solution

sin^-1(sqrt(1-cos^2 theta))

sin theta= opposite/hypotenuse

And from the given fugure, opposite= sqrt(1-cos^2 theta)

and hypotenuse=1

Therefore sin theta= sqrt(1-cos^2 theta)/1= sqrt(1-cos^2 theta)

taking inverse on both sides we get

theta= sin^-1(sqrt(1-cos^2 theta))