Geometry (B): READ: Tangent Ratio Reading

Tangent Ratio

Learning Objectives

Identify the different parts of right triangles.
Identify and use the tangent ratio in a right triangle.
Identify complementary angles in right triangles.
Understand tangent ratios in special right triangles.

Introduction

Now that you are familiar with right triangles, the ratios that relate the sides, as well as other important applications, it is time to learn about trigonometric ratios. Trigonometric ratios show the relationship between the sides of a triangle and the angles inside of it. This lesson focuses on the tangent ratio.

Parts of a Triangle

In trigonometry, there are a number of different labels attributed to different sides of a right triangle. They are usually in reference to a specific angle. The hypotenuse of a triangle is always the same, but the terms adjacent and opposite depend on which angle you are referencing. A side adjacent to an angle is the leg of the triangle that helps form the angle. A side opposite to an angle is the leg of the triangle that does not help form the angle.

In the triangle shown above, segment $\overline{AB}$ is adjacent to $\angle{B}$ , and segment $\overline{AC}$ is opposite to $\angle{B}$ . Similarly, $\overline{AC}$ is adjacent to $\angle{x}$ , and $\overline{AB}$ is opposite $\angle{C}$ . The hypotenuse is always $\overline{BC}$ .

Example 1

Examine the triangle in the diagram below.

Identify which leg is adjacent to $\angle{R}$ , opposite to $\angle{R}$ , and the hypotenuse.

The first part of the question asks you to identify the leg adjacent to $\angle{R}$ . Since an adjacent leg is the one that helps to form the angle and is not the hypotenuse, it must be $\overline{QR}$ . The next part of the question asks you to identify the leg opposite $\angle{R}$ . Since an opposite leg is the leg that does not help to form the angle, it must be $\overline{QS}$ . The hypotenuse is always opposite the right angle, so in this triangle the hypotenuse is segment $\overline{RS}$ .

The Tangent Ratio

The first ratio to examine when studying right triangles is the tangent. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. The hypotenuse is not involved in the tangent at all. Be sure when you find a tangent that you find the opposite and adjacent sides relative to the angle in question.

For an acute angle measuring $x$ , we define $\tan x = \frac{\text{opposite}}{\text{adjacent}}$ .

Example 2

What are the tangents of $\angle{X}$ and $\angle{Y}$ in the triangle below?

To find these ratios, first identify the sides opposite and adjacent to each angle.

$\tan \angle{X} & = \frac{\text{opposite}} {\text{adjacent}} = \frac{5} {12} \approx{0.417} \\ \tan \angle{Y} & = \frac{\text{opposite}} {\text{adjacent}} = \frac{12} {5} = 2.4$

So, the tangent of $\angle{X}$ is about $0.417$ and the tangent of $\angle{Y}$ is $2.4$ .

It is common to write $\tan X$ instead of $\tan \angle X$ . In this text we will use both notations.

Complementary Angles in Right Triangles

Recall that in all triangles, the sum of the measures of all angles must be $180^\circ$ . Since a right angle has a measure of $90^\circ$ , the remaining two angles in a right triangle must be complementary. Complementary angles have a sum of $90^\circ$ . This means that if you know the measure of one of the smaller angles in a right triangle, you can easily find the measure of the other. Subtract the known angle from $90^\circ$ and you’ll have the measure of the other angle.

Example 3

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

To find $m\angle{N}$ , you can subtract the measure of $\angle{N}$ from $90^\circ$ .

$m\angle{N} + m\angle{O} &= 90 \\ m\angle{N} &= 90 - m\angle{O}\\ m\angle{N} &= 90 - 27\\ m\angle{N} &= 63$

So, the measure of $\angle{N}$ is $63^\circ$ since $\angle{N}$ and $\angle{O}$ are complementary.

Tangents of Special Right Triangles

It may help you to learn some of the most common values for tangent ratios. The table below shows you values for angles in special right triangles.

	$30^\circ$	$45^\circ$	$60^\circ$
Tangent	$\frac{1}{\sqrt{3}} \approx 0.577$	$\frac{1}{1} = 1$	$\frac{\sqrt{3}}{1}\approx 1.732$

Notice that you can derive these ratios from the $30^\circ-60^\circ-90^\circ$ special right triangle. You can use these ratios to identify angles in a triangle. Work backwards from the ratio. If the ratio equals one of these values, you can identify the measurement of the angle.

Example 4

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

Find the tangent of $\angle{J}$ and compare it to the values in the table above.

$\tan{J} &= \frac{\text{opposite}} {\text{adjacent}}\\ &= \frac{5} {5}\\ &= 1$

So, the tangent of $\angle{J}$ is $1$ . If you look in the table, you can see that an angle that measures $45^\circ$ has a tangent of $1$ . So, $m \angle{J}=45^\circ$ .

Example 5

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

Find the tangent of $\angle{Z}$ and compare it to the values in the table above.

$\tan{Z} &= \frac{\text{opposite}} {\text{adjacent}}\\ &= \frac{5.2} {3}\\ &= 1.7\bar{3}$

So, the tangent of $\angle{Z}$ is about $1.73$ . If you look in the table, you can see that an angle that measures $60^\circ$ has a tangent of $1.732$ . So, $m\angle{z} \approx 60^\circ$ .

Notice in this example that $\triangle{XYZ}$ is a $30^\circ-60^\circ-90^\circ$ triangle. You can use this fact to see that $XY = 5.2 \approx 3\sqrt{3}$ .

Lesson Summary

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

How to identify the different parts of right triangles.
How to identify and use the tangent ratio in a right triangle.
How to identify complementary angles in right triangles.
How to understand tangent ratios in special right triangles.

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.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Qusetions

Use the following diagram for exercises 1-5.

How long is the side opposite angle $G?$
How long is the side adjacent to angle $G?$
How long is the hypotenuse?
What is the tangent of $\angle G?$
What is the tangent of $\angle H?$
What is the measure of $\angle C$ in the diagram below?
What is the measure of $\angle H$ in the diagram below?

Use the following diagram for exercises 8-9.
What is the tangent of $\angle R?$
What is the tangent of $\angle S?$
What is the measure of $\angle E$ in the triangle below?