Why do the functions f (x) = sin-1 (x) and g (x) = cos-1 (x) have different ranges?

We Know that

For a function to have an inverse function, it must be one-to-one-that is, it must pass the Horizontal Line Test.

1. On the interval [-pi/2, pi/2], the function y = sin x is increasing

2. On the interval [-pi/2, pi/2], y = sin x takes on its full range of values, [-1, 1]

3. On the interval [-pi/2, pi/2], y = sin x is one-to-one

sin x has an inverse function on this interval [-pi/2, pi/2]

On the restricted domain [-pi/2, pi/2] y = sin x has a unique inverse function called the inverse sine function. f (x) = sin-1 (x)

the range of y=sin x in the domain [-pi/2, pi/2] is [-1,1]

the range of y=sin-1 x in the domain [-1,1] is [-pi/2, pi/2]

1. On the interval [0, pi], the function y = cos x is decreasing

2. On the interval [0, pi], y = cos x takes on its full range of values, [-1, 1]

3. On the interval [0, pi], y = cos x is one-to-one

cos x has an inverse function on this interval [0, pi]

On the restricted domain [0, pi] y = cos x has a unique inverse function called the inverse sine function. f (x) = cos-1 (x)

the range of y=cos x in the domain [0, pi] is [-1,1]

the range of y=cos-1 x in the domain [-1,1] is [0, pi]

the answer is

the values of the range are different because the domain in which the inverse function exists are different