Inverse Trigonometric Functions M 140 Precalculus V. J. Motto

Let us remember our discussion on inverse functions: We found inverses for functions by 1.Reversing ordered pairs: (x, y)  (y, x) 2.Reflection the function f across the line y = x 3.Showing that (fog)(x) = x. Then g = f -1. Although every function has an inverse. Not every inverse is a function. For a function to an inverse that is function the function must be one-to-one.

Consider the function f(x) = x 2 : Pictured at the left is the function and its inverse. 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.

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.

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

Finding the inverse by transformations: Rotate 90 o to Left Now reflect on y-axis Combine the graphs! We can use this technique to find our inverses of the trigonometric functions

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?

Take a look at the piece of the graph in the red frame. We are going to build the inverse function from this section of the sine curve because: This section picks up all the outputs of the sine from –1 to 1. 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.

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

The inverse of the sine function by transformations. Rotate 90 o to Left Now reflect on y-axis The resulting graph is the graph of sin -1 x. Sometimes this function is called the arcsine x or arcsin x.

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

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

In the tradition of inverse functions then we have: 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. Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians). Can you determine the degree equivalents?

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.

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. Since the domain and range for the section are the domain and range for the inverse cosine are The transformation is shown on the next slide

Transforming the cosine function into its inverse The resulting graph is the graph of cos -1 x. Sometimes this function is called the arccosine x or arccos x. Rotate 90 o to the left Reflect across the y-axis

The other trig functions require similar restrictions on their domains in order to generate an inverse. Like the sine function, the domain of the section of the tangent that generates the arctan is y=tan(x) y=arctan(x)

Transforming the tan function into its inverse The resulting graph is the graph of tan -1 x. Sometimes this function is called the arctangent x or arctan x. Rotate 90 o to the left Reflect across the y-axis

The table below will summarize 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=arccos(x) will be in which quadrant? When x< 0, y=arcsin(x) will be in which quadrant? When x< 0, y=arctan(x) will be in which quadrant? y<0 in IV y>0 in II y<0 in IV

Using our calculator to evaluate the inverse trigonometric functions On most calculators, you access the inverse trig functions by using the 2 nd function option on the corresponding trig functions. The mode button allows you to choose whether your work will be in degrees or in radians. You have to stay on top of this because the answer is not in a format that tells you which mode you are in.

First, some calculator problems. We will work these problems together to assure that you all learn how to use your calculator.

Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians. Work out the answers yourself before you click.

We can evaluate some inverse trigonometric functions by hand using our special triangles. The next few slides contain some examples.

1 2 1 The graphs give you the big picture concerning the behavior of the inverse trig functions. Calculators are helpful with calculations (later for that). But special triangles can be very helpful with respect to the basics. Use the special triangles above to answer the following. Try to figure it out yourself before you click.

1 2 1 OK, lets try a few more. Try them before you peek.

1 2 Negative inputs for the arccos can be a little tricky. From the triangle you can see that arccos(1/2) = 60 degrees. But negative inputs for the arccos generate angles in Quadrant II so we have to use 60 degrees as a reference angle in the second quadrant. y x 2

You should be able to do inverse trig calculations without a calculator when special angles from the special triangles are involved. You should also be able to do inverse trig calculations without a calculator for quadrantal angles. Its not that bad. Quadrantal angles are the angles between the quadrants—angles like To solve arccos(-1) for example, you could draw a quick sketch of the cosine section: And observe that arccos(-1) =

But a lot of people feel comfortable using the following sketch and the definitions of the trig ratios. For arccos(-1) for example, you can observe that, since the point (-1, 0) is the one we want. That point is on the terminal side of Or for arccot(0), you can observe that, since the point (0, 1) is the one we want. That point is on the terminal side of 90 degrees. So, since So, arccot(0) Good luck getting that answer off of a calculator.

Finally, we encounter the composition of trig functions with inverse trig functions. The following are pretty straightforward compositions. Try them yourself before you click to the answer. First, what do we know about We know that is an angle whose sine is so Did you suspect from the beginning that this was the answer because that is the way inverse functions are SUPPOSED to behave? If so, good instincts but….

Consider a slightly different setup: This is also the composition of two inverse functions but… Did you suspect the answer was going to be 120 degrees? This problem behaved differently because the first angle, 120 degrees, was outside the range of the arcsin. So use some caution when evaluating the composition of inverse trig functions.