Geometry (B): READ: Inverse Trig Reading

Inverse Trigonometric Ratios

Learning Objectives

Identify and use the arctangent ratio in a right triangle.
Identify and use the arcsine ratio in a right triangle.
Identify and use the arccosine ratio in a right triangle.
Understand the general trends of trigonometric ratios.

Introduction

The word inverse is probably familiar to you often in mathematics, after you learn to do an operation, you also learn how to “undo” it. Doing the inverse of an operation is a way to undo the original operation. For example, you may remember that addition and subtraction are considered inverse operations. Multiplication and division are also inverse operations. In algebra you used inverse operations to solve equations and inequalities. You may also remember the term additive inverse, or a number that can be added to the original to yield a sum of $0$ . For example, $5$ and $-5$ are additive inverses because $5 + (-5) = 0$ .

In this lesson you will learn to use the inverse operations of the trigonometric functions you have studied thus far. You can use inverse trigonometric functions to find the measures of angles when you know the lengths of the sides in a right triangle.

Inverse Tangent

When you find the inverse of a trigonometric function, you put the word arc in front of it. So, the inverse of a tangent is called the arctangent (or arctan for short). Think of the arctangent as a tool you can use like any other inverse operation when solving a problem. If tangent tells you the ratio of the lengths of the sides opposite and adjacent to an angle, then arctan tells you the measure of an angle with a given ratio.

Suppose $\tan{X} = 0.65$ . The arctangent can be used to find the measure of $\angle{X}$ on the left side of the equation.

$\arctan{(\tan{X})} & = \arctan{(0.65)} \\ m\angle{X} & = \arctan{(0.65)} \approx 33^\circ$

Where did that $33^\circ$ come from? There are two basic ways to find an arctangent. Sometimes you will be given a table of trigonometric values and the angles to which they correspond. In this scenario, find the value that is closest to the one provided, and identify the corresponding angle.

Another, easier way of finding the arctangent is to use a calculator. The arctangent button may be labeled “arctan,” “atan,” or “ $\tan^{-1}$ .” Either way, select this button, and input the value in question. In this case, you would press the arctangent button and enter $0.65$ (or on some calculators, enter $.65$ , then press “arctan”). The output will be the value of measure $\angle{X}$ .

$m\angle{X} & = \arctan{(0.65)}\\ m\angle{X} & \approx 33$

$m\angle{X}$ is about $33^\circ$ .

Example 1

Solve for $m\angle{Y}$ if $\tan{Y} = 0.384$

You can use the inverse of tangent, arctangent to find this value.

$\arctan{(\tan {Y})} & = \arctan{(0.384)} \\ m\angle{Y} & = \arctan{(0.384)}$

Then use your calculator to find the arctangent of $0.384$ .

$m\angle{Y} \approx 21^\circ$

Example 2

What is $m\angle{B}$ in the triangle below?

First identify the proper trigonometric ratio related to $\angle{B}$ that can be found using the sides given. The tangent uses the opposite and adjacent sides, so it will be relevant here.

$\tan{B} &= \frac{\text{opposite}} {\text{adjacent}}\\ &= \frac{8} {5}\\ &= 1.6$

Now use the arctangent to solve for the measure of $\angle{B}$ .

$\arctan{(\tan{B})} & = \arctan{(1.6)} \\ m\angle{B} & = \arctan{(1.6)}$

Then use your calculator to find the arctangent of $1.6$ .

$m\angle{B} \approx 58^\circ$

Inverse Sine

Just as you used arctangent as the inverse operation for tangent, you can also use arcsine (shortened as arcsin) as the inverse operation for sine. The same rules apply. You can use it to isolate a variable for an angle measurement, but you must perform the operation on both sides of the equation. When you know the arcsine value, use a table or a calculator to find the measure of the angle.

Example 3

Solve for $m\angle{P}$ if $\sin{P} = 0.891$

You can use the inverse of sine, arcsine to find this value.

$\arcsin{(\sin{P})} & = \arcsin{(0.891)} \\ m\angle{P} & = \arcsin{(0.891)}$

Then use your calculator to find the arcsine of $0.891$ .

$m\angle{P} \approx 63^\circ$

Example 4

What is $m \angle{F}$ in the triangle below?

First identify the proper trigonometric ratio related to angle $F$ that can be found using the sides given. The sine uses the opposite side and the hypotenuse, so it will be relevant here.

$\sin{F} & = \frac{\text{opposite}} {\text{adjacent}} \\ \sin{F} & = \frac{12} {13} \\ \sin{F} & \approx 0.923$

Now use the arcsine to isolate the value of angle $F$ .

$\arcsin{(\sin{F})} & = \arcsin{(0.923)} \\ m\angle{F} & = \arcsin{(0.923)}$

Finally, use your calculator to find the arcsine of $0.923$ .

$m\angle{F} \approx 67^\circ$

Inverse Cosine

The last inverse trigonometric ratio is arccosine (often shortened to arccos). The same rules apply for arccosine as apply for all other inverse trigonometric functions. You can use it to isolate a variable for an angle measurement, but you must perform the operation on both sides of the equation. When you know the arccosine value, use a table or a calculator to find the measure of the angle.

Example 5

Solve for $m\angle{Z}$ if $\cos{Z} = 0.31$ .

You can use the inverse of cosine, arccosine, to find this value.

$\arccos{(\cos{Z})} & = \arccos{(0.31)} \\ m\angle{Z} & = \arccos{(0.31)}$

Then use your calculator to find the arccosine of $0.31$ .

$m\angle{Z} \approx 72^\circ$

Example 6

What is the measure of $\angle{K}$ in the triangle below?

First identify the proper trigonometric ratio related to $\angle{K}$ that can be found using the sides given. The cosine uses the adjacent side and the hypotenuse, so it will be relevant here.

$\cos{K} &= \frac{\text{adjacent}} {\text{hypotenuse}}\\ &= \frac{9} {11}\\ &= 0.818$

Now use the arccosine to isolate the value of $\angle{K}$ .

$\arccos{(\cos{K})} & = \arccos{(0.818)} \\ m\angle{K} & = \arccos{(0.818)}$

Finally use your calculator or a table to find the arccosine of $0.818.$

$m\angle{K} \approx 35^\circ$

General Trends in Trigonometric Ratios

Now that you know how to find the trigonometric ratios as well as their inverses, it is helpful to look at trends in the different values. Remember that each ratio will have a constant value for a specific angle. In any right triangle, the sine of a $30^\circ$ angle will always be $0.5$ —it doesn’t matter how long the sides are. You can use that information to find missing lengths in triangles where you know the angles, or to identify the measure of an angle if you know two of the sides.

Examine the table below for trends. It shows the sine, cosine, and tangent values for eight different angle measures.

	$10^\circ$	$20^\circ$	$30^\circ$	$40^\circ$	$50^\circ$	$60^\circ$	$70^\circ$	$80^\circ$
Sine	$0.174$	$0.342$	$0.5$	$0.643$	$0.766$	$0.866$	$0.940$	$0.985$
Cosine	$0.985$	$0.940$	$0.866$	$0.766$	$0.643$	$0.5$	$0.342$	$0.174$
Tangent	$0.176$	$0.364$	$0.577$	$0.839$	$1.192$	$1.732$	$2.747$	$5.671$

Example 7

Using the table above, which value would you expect to be greater: the sine of $25^\circ$ or the cosine of $25^\circ$ ?

You can use the information in the table to solve this problem. The sine of $20^\circ$ is $0.342$ and the sine of $30^\circ$ is $0.5$ . So, the sine of $25^\circ$ will be between the values $0.342$ and $0.5$ . The cosine of $20^\circ$ is $0.940$ and the cosine of $30^\circ$ is $0.866.$ So, the cosine of $25^\circ$ will be between the values of $0.866$ and $0.940.$ Since the range for the cosine is greater, than the range for the sine, it can be assumed that the cosine of $25^\circ$ will be greater than the sine of $25^\circ.$

Notice that as the angle measures approach $90^\circ$ , $\sin$ approaches $1$ . Similarly, as the value of the angles approach $90^\circ$ , the $\cos$ approaches $0$ . In other words, as the $\sin$ gets greater, the $\cos$ gets smaller for the angles in this table.

The tangent, on the other hand, increases rapidly from a small value to a large value (infinity, in fact) as the angle approaches $90^\circ$ .

Lesson Summary

In this lesson, we explored how to work with different radicals both in theory and in practical situations. Specifically, we have learned:

How to identify and use the arctangent ratio in a right triangle.
How to identify and use the arcsine ratio in a right triangle.
How to identify and use the arccosine ratio in a right triangle.
How to understand the general trends of trigonometric ratios.

These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.

Points to Consider

To this point, all of the trigonometric ratios you have studied have dealt exclusively with right triangles. Can you think of a way to use trigonometry on triangles that are acute or obtuse?

Last modified: Tuesday, June 29, 2010, 10:37 AM