Inverse Trigonometric Functions Recall some facts about inverse functions: 1.For a function to have an inverse it must be a one-to-one function. 2.The domain of f -1 is the range of f. 3.The range of f -1 is the domain of f. 4.If ( a,b ) is a point on the graph of f, then ( b,a ) is a point on the graph of f f(f -1 ) = x for all x in the domain of f f -1 (f ) = x for all x in the domain of f.

The graph of y = sin x is given below. Since y = sin x is not a 1-1 function on its domain of all real numbers, it does not have an inverse. Note that it fails the horizontal line test.

We must restrict the domain of the sine function so that it will be a 1-1 function, and have an inverse. This could be done in several ways, but the following is most common:

Definition: Inverse Sine

Another way of writing the inverse of the sine function is as follows:

The graph of y = cos x is given below. Since y = cos x is not a 1-1 function on its domain of all real numbers, it does not have an inverse. Note that it fails the horizontal line test.

We must restrict the domain of the cosine function so that it will be a 1-1 function, and have an inverse. This could be done in several ways, but the following is most common:

Definition: Inverse Cosine

Another way of writing the inverse of the cosine function is as follows:

The graph of y = tan x is given below. Since y = tan x is not a 1-1 function on its normal domain, it does not have an inverse. Note that it fails the horizontal line test.

We must restrict the domain of the tangent function so that it will be a 1-1 function, and have an inverse. This could be done in several ways, but the following is most common:

Definition: Inverse Tangent

Another way of writing the inverse of the tangent function is as follows: