Do you know how to identify a cosine? Do you know what a cosine is?

Knowing about cosines can help you when working with the relationships between side lengths and angle measures in a right triangle. When finished with this Concept, you will understand how to identify a cosine.

### Guidance

One way to analyze right triangles is through ** trigonometric ratios**.

**There are three trigonometric ratios and they help us to understand the proportions between the sides and the angles.**

Pay special attention to the specifics of each ratio, as you will have to remember these.

Let's talk about cosines.

**The cosine is the ratio of the adjacent side of an angle to the hypotenuse.**

Now let's apply this definition to a situation.

**What are the cosines of \begin{align*}\angle M\end{align*} and \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 that an adjacent side is the one that does create the angle and is not the hypotenuse.**

\begin{align*}\text{cosine} \angle M & = \frac{adjacent}{hypotenuse} = \frac{15}{17} \approx 0.88\\ \text{cosine} \angle N & = \frac{adjacent}{hypotenuse} = \frac{8}{17} \approx 0.47\end{align*}

**Once again notice that we divided the numerator by the denominator to find a decimal representation of the cosine of each of the angles. You can figure these ratios out on your calculator.**

**The cosine of \begin{align*}\angle M\end{align*} is about 0.88 and the cosine of \begin{align*}\angle N\end{align*} is about 0.47.**

Use this triangle to answer the following questions. You may round to the nearest hundredth.

The length of the hypotenuse is 13.

#### Example A

What is the cosine of \begin{align*}\angle A\end{align*}?

**Solution: \begin{align*}\frac{5}{13} = .38\end{align*}**

#### Example B

What is the cosine of \begin{align*}\angle B\end{align*}?

**Solution: \begin{align*}\frac{12}{13} = .92\end{align*}**

#### Example C

What is the ratio for cosine?

**Solution: Adjacent side of the identified angle over the hypotenuse.**

Now let's go back to the dilemma from the beginning of the Concept.

A cosine is a trigonometric ratio. When working with a cosine, first there is an angle identified. Then the side length adjacent to that angle is written over the length of the hypotenuse. Dividing the numerator by the denominator will give you a numerical value for the cosine.

### Vocabulary

- Trigonometric Ratios
- ratios that help us to understand the relationships between sides and angles of right triangles.

- Cosine
- the ratio of the adjacent side to the hypotenuse.

### Guided Practice

Here is one for you to try on your own.

Find the cosines of \begin{align*}\angle A\end{align*} and \begin{align*}\angle C\end{align*} in the triangle pictured below.

**Solution**

The cosine is the ratio of the adjacent side to the hypotenuse. Here are the cosine ratios.

\begin{align*}\text{Cosine} \angle A & = \frac{6}{10} = .6\\ \text{Cosine} \angle C &= \frac{8}{10} = .8\end{align*}

### Video Review

Khan Academy Basic Trigonometry

### Practice

Directions: Solve each problem.

- What is the cosine of \begin{align*}\angle G\end{align*}?
- What is the cosine of \begin{align*}\angle H\end{align*}?
- How do you identify a cosine?

- What is the cosine of \begin{align*}\angle R\end{align*}?
- What is the cosine of \begin{align*}\angle S\end{align*}?

- What is the cosine of \begin{align*}\angle A\end{align*}?
- What is the cosine of \begin{align*}\angle B\end{align*}?
- What is the length of the missing side rounded to the nearest hundredth?

Directions: Answer each question.

9. Is the cosine related to an angle?

10. Do you need to know side lengths of a triangle to write a cosine?

11. Which side lengths do you need?

12. If the cosine was \begin{align*}\frac{5}{20}\end{align*}, what would the numerical value of the cosine be?

13. If the cosine was \begin{align*}\frac{5}{25}\end{align*}, what would the numerical value of the cosine be?

14. If the cosine was \begin{align*}\frac{3}{33}\end{align*}, what would the numerical value of the cosine be?

15. If the cosine was \begin{align*}\frac{12}{14}\end{align*}, what would the numerical value of the cosine be?