# 2.1.7: TAN

- Page ID
- 4218

Explore the tangent ratio and use it to find missing sides of right triangles.

As the measure of an angle increases between \(0^{\circ}\) and \(90^{\circ}\), how does the **tangent** ratio of the angle change?

**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. \(\overline{DF}\) and \(\overline{AC}\) are corresponding sides because they are both opposite the \(22^{\circ}\) angle. \(\dfrac{DF}{AC}=\dfrac{4}{2}=2\), so the scale factor between the two triangles is 2. This means that \(x=10\), because \(\dfrac{FE}{CB}=\dfrac{10}{5}=2\).

The ratio between the two legs of any \(22^{\circ}\) right triangle will always be the same, because all \(22^{\circ}\) right triangles are similar. The ratio of the ** length of the leg opposite** the \(22^{\circ}\) angle to the

**length of the leg adjacent****the \(22^{\circ}\) angle will be \(\dfrac{2}{5}=0.4\). You can use this fact to find a missing side of another \(22^{\circ}\) right triangle.**

**to**Because this is a \(22^{\circ}\) right triangle, you know that \(\dfrac{\text{opposite leg}}{\text{adjacent leg}}=\dfrac{2}{5}=0.4\).

\(\begin{aligned} \dfrac{\text{opposite leg}}{\text{adjacent leg}}&=0.4 \\ \dfrac{7}{x}&=0.4 \\ 0.4x &=7\\ x&=17.5 \end{aligned}\)

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

**programmed into it, so that you can determine the \dfrac{\text{opposite leg}}{\text{adjacent leg}}\) ratio for any angle within a right triangle. The abbreviation for**

**tangent****is**

**tangent****.**

**tan****Calculating Tangent Functions**

Use your calculator to find the tangent of \(75^{\circ}\). What does this value represent?

Make sure your calculator is in degree mode. Then, type “\(\tan(75)\)”.

\(tan(75^{\circ})\approx 3.732\)

This means that the ratio of the **length of the opposite leg to the length of the adjacent leg** for a \(75^{\circ}\) angle within a right triangle will be approximately 3.732.

**Solving for Unknown Values **

1. Solve for \(x\).

From the previous problem, you know that the ratio \(\dfrac{\text{opposite leg}}{\text{adjacent leg}} \approx 3.732\). You can use this to solve for \(x\).

\(\begin{aligned}

\dfrac{\text { opposite leg }}{\text { adjacent leg }} & \approx 3.732 \\

\dfrac{x}{2} & \approx 3.732 \\

x & \approx 7.464

\end{aligned}\)

2. Solve for \(x\) and \(y\).

You can use the \(65^{\circ}\) angle to find the correct ratio between 24 and \(x\).

\(\begin{aligned}

\tan \left(65^{\circ}\right) &=\dfrac{\text { opposite leg }}{\text { adjacent leg }} \\

2.145 & \approx \dfrac{24}{x} \\

x & \approx \dfrac{24}{2.145} \\

x & \approx 11.189

\end{aligned}\)

Note that this answer is only approximate because you rounded the value of \(\tan 65^{\circ}\). An exact answer will include “tan”. The exact answer is:

\(x=\dfrac{24}{\tan 65^{\circ}}\)

To solve for y, you can use the Pythagorean Theorem because this is a right triangle.

\(\begin{array}{r}

11.189^{2}+24^{2}=y^{2} \\

701.194=y^{2} \\

26.48=y

\end{array}\)

Example \(\PageIndex{1}\)

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

**Solution**

As the measure of an angle increases between \(0^{\circ}\) and \(90^{\circ}\), how does the tangent ratio of the angle change?

As an angle increases, the length of its opposite leg increases. Therefore, \(\dfrac{\text{opposite leg}}{\text{adjacent leg}}\) increases and thus the value of the tangent ratio increases.

Example \(\PageIndex{2}\)

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?

**Solution**

Two right triangles with a \(32^{\circ}\) angle will be similar. Two non-right triangles with a \(32^{\circ}\) angle will not necessarily be similar. The tangent ratio works for right triangles because all right triangles with a given angle are similar. The tangent ratio doesn't work in the same way for non-right triangles because not all non-right triangles with a given angle are similar. **You can only use the tangent ratio for right triangles.**** **

Example \(\PageIndex{3}\)

Use your calculator to find the tangent of \(45^{\circ}\). What does this value represent? Why does this value make sense?

**Solution**

\(\tan(45^{\circ})=1\). This means that the ratio of the length of the opposite leg to the length of the adjacent leg is equal to 1 for right triangles with a \(45^{\circ}\) angle.

This should make sense because right triangles with a \(45^{\circ}\) angle are isosceles. The legs of an isosceles triangle are congruent, so the ratio between them will be 1.

Example \(\PageIndex{4}\)

Solve for \(x\).

**Solution**

Use the tangent ratio of a \(35^{\circ}\) angle.

\(\begin{aligned}

\tan \left(35^{\circ}\right) &=\dfrac{\text { opposite leg }}{\text { adjacent leg }} \\

\tan \left(35^{\circ}\right) &=\dfrac{x}{18} \\

x &=18 \tan \left(35^{\circ}\right) \\

x & \approx 12.604

\end{aligned}\)

**Review**

1. Why are all right triangles with a \(40^{\circ}\) angle similar? What does this have to do with the tangent ratio?

2. Find the tangent of \(40^{\circ}\).

3. Solve for \(x\).

4. Find the tangent of \(80^{\circ}\).

5. Solve for \(x\).

6. Find the tangent of \(10^{\circ}\).

7. Solve for \(x\).

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

9. Find the tangent of \(27^{\circ}\).

10. Solve for \(x\).

11. Find the tangent of \(42^{\circ}\).

12. Solve for \(x\).

13. A right triangle has a \(42^{\circ}\) angle. The base of the triangle, adjacent to the \(42^{\circ}\) 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 \(1:\sqrt{3}:2\). Find the tangent of \(30^{\circ}\). 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 \(0^{\circ}\) to \(90^{\circ}\).

**Review (Answers)**

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

## Vocabulary

Term | Definition |
---|---|

AA Similarity Postulate |
If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar. |

Congruent |
Congruent figures are identical in size, shape and measure. |

Similar |
Two figures are similar if they have the same shape, but not necessarily the same size. |

Tangent |
The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle. |