Geometry (B): READ: Sine and Cosine Reading

Sine and Cosine Ratios

Learning Objectives

Review the different parts of right triangles.
Identify and use the sine ratio in a right triangle.
Identify and use the cosine ratio in a right triangle.
Understand sine and cosine ratios in special right triangles.

Introduction

Now that you have some experience with tangent ratios in right triangles, there are two other basic types of trigonometric ratios to explore. The sine and cosine ratios relate opposite and adjacent sides of a triangle to the hypotenuse. Using these three ratios and a calculator or a table of trigonometric ratios you can solve a wide variety of problems!

Review: Parts of a Triangle

The sine and cosine ratios relate opposite and adjacent sides to the hypotenuse. You already learned these terms in the previous lesson, but they are important to review and commit to memory. The hypotenuse of a triangle is always opposite the right angle, 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.

Example 1

Examine the triangle in the diagram below.

Identify which leg is adjacent to angle $N$ , which leg is opposite to angle $N$ , and which segment is the hypotenuse.

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

The Sine Ratio

Another important trigonometric ratio is sine. A sine ratio must always refer to a particular angle in a right triangle. The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Remember that in a ratio, you list the first item on top of the fraction and the second item on the bottom.

So, the ratio of the sine will be

$\sin x = \frac{\text{opposite}} {\text{hypotenuse}}.$

Example 2

What are $\sin A$ and $\sin B$ in the triangle below?

All you have to do to find the solution is build the ratio carefully.

$\sin A & = \frac{\text{opposite}} {\text{hypotenuse}} = \frac{3} {5} = 0.6 \\ \sin B & = \frac{\text{opposite}} {\text{hypotenuse}} = \frac{4} {5} = 0.8$

So, $\sin A = 0.6$ and $\sin B = 0.8$ .

The Cosine Ratio

The next ratio to examine is called the cosine. The cosine is the ratio of the adjacent side of an angle to the hypotenuse. Use the same techniques you used to find sines to find cosines.

$\cos(\text{angle}) = \frac{\text{adjacent}} {\text{hypotenuse}}$

Example 3

What are the cosines of $\angle{M}$ and $\angle{N}$ in the triangle below?

To find these ratios, identify the sides adjacent to each angle and the hypotenuse. Remember that an adjacent side is the one that does create the angle and is not the hypotenuse.

$\cos M & = \frac{\text{adjacent}} {\text{hypotenuse}} = \frac{15} {17} \approx 0.88 \\ \cos N & = \frac{\text{adjacent}} {\text{hypotenuse}} = \frac{8} {17} \approx 0.47$

So, the cosine of $\angle{M}$ is about $0.88$ and the cosine of $\angle{N}$ is about $0.47$ .

Note that $\triangle{LMN}$ is NOT one of the special right triangles, but it is a right triangle whose sides are a Pythagorean triple.

Sines and Cosines of Special Right Triangles

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

	$30^\circ$	$45^\circ$	$60^\circ$
Sine	$\frac{1}{2}=0.5$	$\frac{1}{\sqrt{2}}\approx 0.707$	$\frac{\sqrt{3}}{2}\approx 0.866$
Cosine	$\frac{\sqrt{3}}{2}\approx 0.866$	$\frac{1}{\sqrt{2}}\approx 0.707$	$\frac{1}{2}=0.5$

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 the measure of $\angle{C}$ in the triangle below?

Note: Figure is not to scale.

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

$\sin{C} &= \frac{\text{opposite}} {\text{hypotenuse}}\\ &= \frac{12} {24}\\ &= 0.5$

So, the sine of $\angle{C}$ is $0.5$ . If you look in the table, you can see that an angle that measures $30^\circ$ has a sine of $0.5$ . So, $m\angle{C}= 30^\circ$ .

Example 5

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

Find the cosine of $\angle{G}$ and compare it to the values in the previous table.

$\cos{G} &= \frac{\text{adjacent}} {\text{hypotenuse}}\\ &= \frac{3} {4.24}\\ &= 0.708$

So, the cosine of $\angle{G}$ is about $0.708$ . If you look in the table, you can see that an angle that measures $45^\circ$ has a cosine of $0.707$ . So, $\angle{G}$ measures about $45^\circ$ . This is a $45^\circ-45^\circ-90^\circ$ right triangle.

Lesson Summary

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

The different parts of right triangles.
How to identify and use the sine ratio in a right triangle.
How to identify and use the cosine ratio in a right triangle.
How to apply sine and cosine 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.

Points to Consider

Before you begin the next lesson, think about strategies you could use to simplify an equation that contains a trigonometric function.

Note, you can only use the $\sin , \cos$ , and $\tan$ ratios on the acute angles of a right triangle. For now it only makes sense to talk about the $\sin , \cos$ , or $\tan$ ratio of an acute angle. Later in your mathematics studies you will redefine these ratios in a way that you can talk about $\sin , \cos$ , and $\tan$ of acute, obtuse, and even negative angles.

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

Review Questions

Use the following diagram for exercises 1-3.

What is the sine of $\angle V?$
What is the cosine of $\angle V?$
What is the cosine of $\angle U?$
Use the following diagram for exercises 4-6.
What is the sine of $\angle O?$
What is the cosine of $\angle O?$
What is the sine of $\angle M?$
What is the measure of $\angle H$ in the diagram below?

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