Have you ever built a ramp? Do you know how to use angles and side lengths with ratios? Take a look at this dilemma.

Karen is building a ramp. She has a height, a base and a point drawn in her design. She is wondering what the tangent of the angle would be if the height of the ramp is 2.5 feet and the base is 6 feet? Here is a picture of her ramp drawing. She needs to find the tangent of \begin{align*}\angle B\end{align*}.

Do you know how to figure this out?

**Pay attention and you will learn how to answer this question by the end of the Concept.**

### Guidance

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

**These trigonometric ratios help us to understand the proportions between the sides and the angles.**

Let's look at a trigonometric ratio called a tangent.

**The tangent is the ratio of the opposite side to the adjacent side of an angle.**

The hypotenuse is not involved in the tangent at all.

**What are the tangents of \begin{align*}\angle X\end{align*} and \begin{align*}\angle Y\end{align*} in the triangle below?**

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

\begin{align*}\text{tangent} \angle X & = \frac{opposite}{adjacent} = \frac{5}{12} \approx 0.417\\ \text{tangent} \angle Y & = \frac{opposite}{adjacent} = \frac{12}{5} = 2.4\end{align*}

**The tangent of \begin{align*}\angle X\end{align*} is about 0.417 and the tangent of \begin{align*}\angle Y\end{align*} is 2.4.**

This Concept focuses on tangents, but there are two other trigonometric ratios, the sine and the cosine.

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 tangent of \begin{align*}\angle A\end{align*}?

**Solution: \begin{align*}\frac{12}{5} = 2.4\end{align*}**

#### Example B

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

**Solution: \begin{align*}\frac{5}{12} = .42\end{align*}**

#### Example C

What is the ratio for tangent?

**Solution: Side length opposite the angle over the hypotenuse.**

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

To figure this out, we first have to write the ratio of the length of the side opposite to \begin{align*}\angle B\end{align*} compared with the adjacent side.

\begin{align*}\frac{2.5}{6}\end{align*}

Next we divide.

\begin{align*}0.42\end{align*}

**This is the tangent for \begin{align*}\angle B\end{align*}.**

### Vocabulary

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

- Sine
- the ratio of the opposite side to the hypotenuse.

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

- Tangent
- the ratio of the opposite side to the adjacent.

### Guided Practice

Here is one for you to try on your own.

Find the tangent ratios in the following triangle.

**Solution**

Tangent is the ratio of the opposite side to the adjacent side. Here are the tangent ratios.

\begin{align*}\text{Tangent} \angle A & = \frac{8}{6} = 1.333\\ \text{Tangent} \angle C & = \frac{6}{8} = .75\end{align*}

### Video Review

Khan Academy Basic Trigonometry

### Practice

Directions: solve each problem.

- What is the tangent of \begin{align*}\angle G\end{align*}?
- What is the tangent of \begin{align*}\angle H\end{align*}?
- True or false. The sine and the tangent use the same ratios.

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

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

Directions: Answer each question. You may round to the nearest hundredth.

- If a tangent ratio is \begin{align*}\frac{4}{5}\end{align*}, what is the measure of the tangent?
- If a tangent ratio is \begin{align*}\frac{14}{20}\end{align*}, what is the measure of the tangent?
- If a tangent ratio is \begin{align*}\frac{6}{7}\end{align*}, what is the measure of the tangent?
- If a tangent ratio is \begin{align*}\frac{9}{2}\end{align*}, what is the measure of the tangent?
- If a tangent ratio is \begin{align*}\frac{4}{20}\end{align*}, what is the measure of the tangent?
- If a tangent ratio is \begin{align*}\frac{7}{21}\end{align*}, what is the measure of the tangent?
- If a tangent ratio is \begin{align*}\frac{9}{19}\end{align*}, what is the measure of the tangent?