1. 9. Inverse Trig Functions

2. Previously you have learned.
Previously you have learned
To find an inverse of a function, switch x and y, then solve the equation for y.
 Inverse function notation f¯¹(x)
 For a function to have an inverse it has to be one-to-one. One x for one y value, and one y for one x value. It will pass the vertical and the horizontal line test.
 Two inverse functions on the graph reflect over y=x
f(x)  f¯¹(x) (x,y)  (y, x)

3. Let us begin with a simple question:
What is the first pair of inverse functions that pop into YOUR mind?
This may not be your pair but this is a famous pair. But something is not quite right with this pair. Do you know what is wrong?
Congratulations if you guessed that the top function does not really have an inverse because it is not 1-1 and therefore, the graph will not pass the horizontal line test.

4. Consider the graph of
Note the two points on the graph and also on the line y=4.
f(2) = 4 and f(-2) = 4 so what is an inverse function supposed to do with 4?
By definition, a function cannot generate two different outputs for the same input, so the sad truth is that this function, as is, does not have an inverse.

5. So how is it that we arrange for this function to have an inverse?4
y=x 2
We consider only one half of the graph: x > 0.
The graph now passes the horizontal line test and we do have an inverse:
This process of considering only part of the graph is called RESTRICTING THE DOMAIN
Note how each graph reflects across the line y = x onto its inverse.

6. A similar restriction on the domain is necessary to create an inverse function for each trig function.
Consider the sine function.
You can see right away that the sine function does not pass the horizontal line test.
But we can come up with a valid inverse function if we restrict the domain as we did with the previous function.
How would YOU restrict the domain?

7. Take a look at the piece of the graph in the red frame.
This section is always increasing so it passes the horizontal line test
This section includes the origin. Quadrant I angles generate the positive ratios and negative angles in Quadrant IV generate the negative ratios.
Lets zoom in and look at some key points in this section.

8. I have plotted the special angles on the curve and the table.

9. The new table generates the graph of the inverse.
The domain of the chosen section of the sine is
So the range of the arcsin is
To get a good look at the graph of the inverse function, we will “turn the tables” on the sine function.
The range of the chosen section of the sine is [-1 ,1] so the domain of the arcsin is [-1, 1].

10. Note how each point on the original graph gets “reflected” onto the graph of the inverse.
etc.
You will see the inverse listed as both:

11. Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).
The thing to remember is that for the trig function the input is the angle and the output is the ratio, but for the inverse trig function the input is the ratio and the output is the angle.

12. From Algebra II, remember that an odd function (which the sine is) is symmetric with respect to the origin as can be seen here
This means that plugging in a negative value for x will give you a negative value for y. This is another reason why we choose the 1st and 4th quadrants for sine’s restricted domain.

13. The other inverse trig functions are generated by using similar restrictions on the domain of the trig function. Consider the cosine function:
What do you think would be a good domain restriction for the cosine?
Congratulations if you realized that the restriction we used on the sine is not going to work on the cosine.

14. The chosen section for the cosine is in the red frame
The chosen section for the cosine is in the red frame. This section includes all outputs from –1 to 1 and all inputs in the first and second quadrants. Notice it is always decreasing.
Since the domain and range for the section are the domain and range for the inverse cosine are

15. Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here
This means that plugging in a negative value for x will give you a positive value for y. This is another reason why we have to choose the 1st and 2nd quadrants for cosine’s restricted domain.

16. Like the sine function, the domain of the section of the
tangent that generates the arctan is This is because tan is also an ODD function.
y=arctan(x)
y=tan(x)

17. arcsin(x) arccos(x) arctan(x) Domain Range
The table below summarizes the parameters we have so far. Remember, the angle is the input for a trig function and the ratio is the output. For the inverse trig functions the ratio is the input and the angle is the output.
arcsin(x)
arccos(x)
arctan(x)
Domain
Range
When x<0, y=arcsin(x) will be in which quadrant?
y<0 in IV
When x<0, y=arccos(x) will be in which quadrant?
y>0 in II
y<0 in IV
When x<0, y=arctan(x) will be in which quadrant?

18. Find the exact value of each expression without using a calculator
Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians.