As the measure of an angle increases between

### Tangent Ratio

Recall that one way to show that two triangles are similar is to show that they have two pairs of congruent angles. This means that two right triangles will be similar if they have one pair of congruent non-right angles.

The two right triangles above are similar because they have two pairs of congruent angles. This means that their corresponding sides are proportional.

The ratio between the two legs of any **length of the leg opposite** the **length of the leg adjacent** **to** the

Because this is a

The ratio between the opposite leg and the adjacent leg for a given angle in a right triangle is called the **tangent ratio**. Your scientific or graphing calculator has **tangent** programmed into it, so that you can determine the **tangent** is **tan**.

#### Calculating Tangent Functions

Use your calculator to find the tangent of

Make sure your calculator is in degree mode. Then, type “

This means that the ratio of the ** length of the opposite leg to the length of the adjacent leg** for a

#### Solving for Unknown Values

1. Solve for

From the previous problem, you know that the ratio

2. Solve for

You can use the

Note that this answer is only approximate because you rounded the value of

To solve for

**Examples**

**Example 1**

Earlier, you were asked how does the tangent ratio of the angle change.

As the measure of an angle increases between

As an angle increases, the length of its opposite leg increases. Therefore,

#### Example 2

Tangent tells you the ratio of the two legs of a right triangle with a given angle. Why does the tangent ratio not work in the same way for non-right triangles?

Two right triangles with a **You can only use the tangent ratio for right triangles.**** **

**Example 3**

Use your calculator to find the tangent of

This should make sense because right triangles with a

#### Example 4

Solve for

Use the tangent ratio of a

### Review

1. Why are all right triangles with a

2. Find the tangent of

3. Solve for \begin{align*}x\end{align*}.

4. Find the tangent of \begin{align*}80^\circ\end{align*}.

5. Solve for \begin{align*}x\end{align*}.

6. Find the tangent of \begin{align*}10^\circ\end{align*}.

7. Solve for \begin{align*}x\end{align*}.

8. Your answer to #5 should be the same as your answer to #7. Why?

9. Find the tangent of \begin{align*}27^\circ\end{align*}.

10. Solve for \begin{align*}x\end{align*}.

11. Find the tangent of \begin{align*}42^\circ\end{align*}.

12. Solve for \begin{align*}x\end{align*}.

13. A right triangle has a \begin{align*}42^\circ\end{align*} angle. The base of the triangle, adjacent to the \begin{align*}42^\circ\end{align*} angle, is 5 inches. Find the area of the triangle.

14. Recall that the ratios between the sides of a 30-60-90 triangle are \begin{align*}1:\sqrt{3} : 2\end{align*}. Find the tangent of \begin{align*}30^\circ\end{align*}. Explain how this matches the ratios for a 30-60-90 triangle.

15. Explain why it makes sense that the value of the tangent ratio increases as the angle goes from \begin{align*}0^\circ\end{align*} to \begin{align*}90^\circ\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.1.