Introduction

When we learned about trig functions, we observed that in any right triangle, if we know the measure of one additional angle and the length of one additional side, we can determine all of the other parts of the triangle. With the inverse trigonometric functions that we developed in the last two sections, we are now also able to determine the missing angles in any right triangle where we know the lengths of two sides.

While the original trigonometric functions take a particular angle as input and provide an output that can be viewed as the ratio of two sides of a right triangle, the inverse trigonometric functions take an input that can be viewed as a ratio of two sides of a right triangle and produce the corresponding angle as output. Indeed, it’s imperative to remember that statements such as

say the exact same thing from two different perspectives, and that we read “” as “the angle whose cosine is ”.
Consider a right triangle that has one leg of length and another leg of length . Let be the angle that lies opposite the shorter leg. Sketch a labeled picture of the triangle.
(a)
What is the exact length of the triangle’s hypotenuse?
(b)
What is the exact value of ?
(c)
Rewrite your equation from (b) using the arcsine function in the form , where and are numerical values.
(d)
What special angle from the unit circle is ?

Evaluating Inverse Trigonometric Functions

Like the trigonometric functions themselves, there are a handful of important values of the inverse trigonometric functions that we can determine exactly without the aid of a computer. For instance, we know from the unit circle that , , and . In these evaluations, we have to be careful to remember that the range of the arccosine function is , while the range of the arcsine function is and the range of the arctangent function is , in order to ensure that we choose the appropriate angle that results from the inverse trigonometric function. This is why our emphasis is now turning to the graphs of these functions.

In addition, there are many other values at which we may wish to know the angle that results from an inverse trigonometric function. To determine such values, one can use a computational device (such as Desmos) in order to find an approximation; however, in this class we leave it in the form , as this is the exact value.

We can also use inverse trigonometric functions to solve equations that up until now, have been unsolvable for us.

Using Inverse Trig in Applied Contexts

Now that we have developed the (restricted) sine, cosine, tangent, and secant functions and their respective inverses, in any setting in which we have a right triangle together with one side length and any one additional piece of information (another side length or a non-right angle measurement), we can determine all of the remaining pieces of the triangle. In the example that follows and the homework, we explore these possibilities in a variety of different applied contexts.

On a baseball diamond (which is a square with -foot sides), the third baseman fields the ball right on the line from third base to home plate and feet away from third base (towards home plate). Give exact solutions without using a computational device.
(a)
When he throws the ball to first base, what angle does the line the ball travels make with the first base line?
(b)
What angle does it make with the third base line? Draw a well-labeled diagram.
(c)
What angles arise if he throws the ball to second base instead?
Give exact solutions without using a computational device. A camera is tracking the launch of a SpaceX rocket. The camera is located ’ from the rocket’s launching pad, and the camera elevates in order to keep the rocket in focus.
(a)
At what angle is the camera tilted when the rocket is ’ off the ground?
Now, rather than considering the rocket at a fixed height of ft, let its height vary and call the rocket’s height .
(b)
Determine the camera’s angle, as a function of , and compute the average rate of change of on the intervals , , and .
(c)
What do you observe about how the camera angle is changing?

Further Exploration

When composing trigonometric functions with inverse trigonometric functions, the expressions can often be rewritten as algebraic expressions of . We will see two examples of this below.

A Note on Triangles

We can now use trigonometry to find angles of right triangles if we know the side lengths and side lengths of right triangles if we know the angles. You might be wondering, “What about triangles that are not right triangles? Can we use trig to learn anything about those?” It turns out that the Law of Sines and the Law of Cosines gives use a way to analyze other triangles beyond just right triangles using trig functions. For more information about this topic, see Laws of Sines and Cosines by Katherine Yoshiwara.