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

In this exercise we want to explain why two more similar functions have different ranges, like this:

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

In this exercise we know that we are dealing with two distinct functions, like this:

What is the function of sine?

The sine function is considered an odd function, as there is a symmetry in the graph with respect to the bisector of the odd quadrants. When a function is considered odd, we have that f (x) = -f (x), that is, sin (-x) = -sin (x).

What is the function of cosine?

Cosine is a trigonometric function, used in a right triangle to define the ratio of the side adjacent to and the hypotenuse of this triangle.

So we can see that the reason the two functions have different ranges is associated with them having different domains.

See more about functions at link

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  

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