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5.5: Cosine Ratio

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Identify and use the cosine ratio in a right triangle.
• Understand sine and cosine ratios in special right triangles.

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.

This means that the cosine ratio is: the ____________________ side divided by the _______________________.

cosθ=adjacenthypotenuse\begin{align*}\cos \theta = \frac{adjacent}{hypotenuse}\end{align*}

Use the same techniques you used to find sines to find cosines.

Example 1

What are the cosines of M\begin{align*}\angle{M}\end{align*} and N\begin{align*}\angle{N}\end{align*} in the triangle below?

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

The hypotenuse is the segment ___________, which is _______ cm long.

The side adjacent to angle M\begin{align*}M\end{align*} is the segment ___________, which is _______ cm long.

The side adjacent to angle N\begin{align*}N\end{align*} is the segment ___________, which is _______ cm long.

cosMcosN=adjacenthypotenuse=1517=adjacenthypotenuse=817\begin{align*}\cos M & = \frac {adjacent}{hypotenuse} = \frac {15}{17}\\ \cos N &= \frac {adjacent}{hypotenuse} = \frac {8}{17}\end{align*}

So,the cosine of M\begin{align*}\angle{M}\end{align*} is 1517\begin{align*}\frac{15}{17}\end{align*} and the cosine of N\begin{align*}\angle{N}\end{align*} is 817\begin{align*}\frac{8}{17}\end{align*}.

Note that ΔLMN\begin{align*}\Delta{LMN}\end{align*} on the previous page is NOT one of the special right triangles, but it is a right triangle whose sides are a Pythagorean triple.

1. In the triangle above, which side is the hypotenuse? _____________________

And which side is adjacent to angle Y\begin{align*}Y\end{align*}? _____________________

2. Fill in the blanks and reduce all fractions:

cosY===\begin{align*}\cos Y = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} =\end{align*}

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\begin{align*}30^\circ\end{align*} 45\begin{align*}45^\circ\end{align*} 60\begin{align*}60^\circ\end{align*}
Sine 12\begin{align*}\frac{1}{2}\end{align*} 1222=22\begin{align*}\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\end{align*} 320.866\begin{align*}\frac{\sqrt{3}}{2} \approx 0.866\end{align*}
0.707\begin{align*}\approx 0.707\end{align*}
Cosine 320.866\begin{align*}\frac{\sqrt{3}}{2} \approx 0.866\end{align*} 1222=22\begin{align*}\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\end{align*} 12\begin{align*}\frac{1}{2}\end{align*}
0.707\begin{align*}\approx 0.707\end{align*}

These values, like the tangent values, are derived from the 306090\begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} and the \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} triangles. To find them, use your triangles and set up ratios for sine and cosine using the side lengths. Make sure to simplify your fractions! Some of the decimal values are given as well in case you need them in the following examples.

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 2

What is the measure of \begin{align*}\angle{C}\end{align*} in the triangle below?

Note: Figure is not to scale.

Find the sine of \begin{align*}\angle{C}\end{align*} and compare it to the values in the table above.

Since we are using angle \begin{align*}C\end{align*} and sine, we can see that the opposite side is segment _________, which has a length of ________ cm.

We also know that the hypotenuse in this triangle is segment _________, which has a length of ________ cm.

\begin{align*}\sin C & = \frac{opposite}{hypotenuse}\\ & = \frac{12}{24}\\ (reduce \ !) & = \frac{1}{2}\end{align*}

So, the sine of \begin{align*}\angle{C}\end{align*} is \begin{align*}\frac{1}{2}\end{align*}.

If you look in the table on the previous page, you can see that an angle that measures \begin{align*}30^\circ\end{align*} has a sine of \begin{align*}\frac{1}{2}\end{align*}. So, \begin{align*}m\angle{C}=30^\circ\end{align*}.

(Note: in the table there are two values that equal \begin{align*}\frac{1}{2}\end{align*},but only one of them is for sine! The other value is for cosine, which we do not need in this example.)

Example 3

What is the measure of \begin{align*}\angle{G}\end{align*} in the triangle below?

Find the cosine of \begin{align*}\angle{G}\end{align*} and compare it to the values in the table.

Since we are using angle \begin{align*}G\end{align*} and cosine, we can see that the adjacent side is segment _________, which has a length of ________ cm.

We also know that the hypotenuse in this triangle is segment _________, which has a length of ________ cm.

\begin{align*}\cos \ G & = \frac{adjacent}{hypotenuse}\\ & = \frac{3}{4.24}\\ & = 0.708\end{align*}

So, the cosine of \begin{align*}\angle{G}\end{align*} is about 0.708.

If you look in the table, you can see that an angle that measures \begin{align*}45^\circ\end{align*} has a cosine of 0.707. So, \begin{align*}\angle{G}\end{align*} measures about \begin{align*}45^\circ\end{align*}.

This example is a \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} right triangle.

Below is a \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} triangle.

1. Which side length is the hypotenuse?

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

2. Show your work to find the sine of a \begin{align*}45^\circ\end{align*} angle. (Notice that it does NOT matter which \begin{align*}45^\circ\end{align*} angle you choose!)

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

3. Show your work to find the cosine of a \begin{align*}45^\circ\end{align*} angle. (Notice that it does NOT matter which \begin{align*}45^\circ\end{align*} angle you choose!)

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

4. Are your answers to #2 and #3 above the same as the values in the table?

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

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