Inverse Trigonometric Functions

For a function to have an inverse, it must pass the Horizontal Line Test. As shown below, y=sinx does not pass the test.




The graph is one-to-one only from peak to trough.

Because of this, in order to have an inverse function, we must restrict the domain to the interval:

.

The range would then be [-1,1]



So the restricted domain of , y=sinx has an inverse called the inverse sine function...

The graph would look something like this...



***The output of an inverse trigonometric function is an angle. In this case the output of the inverse sine would be in either the first or the fourth quadrant.***

The cosine function is decreasing on the interval as shown below




As with the sine function, we must also restrict the domain of the cosine function in order to have an inverse cosine function.

The restricted domain would be: [-1,1]
The range would be:

This graph would look like this....

***Again, remember the output of an inverse trigonometric function is an angle. So for this case the output of the inverse cosine function would be in the first or second quadrant.***





Similar to the inverse sine and cosine functions, you can define an inverse tangent function also by restricting the domain.

We restrict the domain of the inverse tangent function to all real numbers.
The range being .

The graph would like like this....




***Don't forget the output of an inverse trigonometric function is an angle. In this case the output of the inverse tangent lies in the first or fourth quadrant.***