5.5: Cosine Ratio
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 _______________________.
Use the same techniques you used to find sines to find cosines.
Example 1
What are the cosines of
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
The side adjacent to angle
So,the cosine of
Note that
Reading Check:
1. In the triangle above, which side is the hypotenuse? _____________________
And which side is adjacent to angle
2. Fill in the blanks and reduce all fractions:
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:





Sine 





Cosine 




These values, like the tangent values, are derived from the
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
Note: Figure is not to scale.
Find the sine of
Since we are using angle
We also know that the hypotenuse in this triangle is segment _________, which has a length of ________ cm.
So, the sine of
If you look in the table on the previous page, you can see that an angle that measures
(Note: in the table there are two values that equal
Example 3
What is the measure of
Find the cosine of
Since we are using angle
We also know that the hypotenuse in this triangle is segment _________, which has a length of ________ cm.
So, the cosine of
If you look in the table, you can see that an angle that measures
This example is a
Reading Check (Challenge):
Below is a
1. Which side length is the hypotenuse?
2. Show your work to find the sine of a
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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!)
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4. Are your answers to #2 and #3 above the same as the values in the table?
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