Have you ever built and launched a rocket? Take a look at this dilemma.

Craig launched a model rocket and wanted to calculate how high it went. He stood 10 feet away from the launch site and used a tool to calculate the angle between the ground and his line of sight to the rocket at its highest point. His data is shown in the diagram below.

Calculate the greatest height the model rocket reached.

**To figure this out, you will need to use a trigonometric ratio called a tangent. Pay attention and you will learn how to successfully solve this problem.**

### Guidance

**Let's think about trigonometric ratios.**

A trigonometric ratio for a specific angle will remain constant no matter how large or small the triangle is. The idea is that the sides will always be in proportion to each other. So, if you know the measure of an angle (and can therefore identify the value of a trigonometric ratio) and the value of one side, you can use trigonometry to calculate the lengths of other sides.

The trick is to use good algebra technique, and make sure that every time you set up a ratio, you are putting the values and variables in the correct places.

**You can find trigonometric ratios by using your calculator.**

You understand trigonometric ratios and have had a chance to practice reading specific values out of a table.

**You can find the ratio for any trigonometric value using your calculator. Take a moment to locate the buttons for sine, cosine, and tangent on the calculator. Keep in mind that usually, sine is abbreviated as** *sin***, cosine is usually abbreviated as** *cos***, and tangent is usually abbreviated as** *tan.*

Press the key of the ratio you want to find, and enter the angle in question. If you hit enter, or calculate, the calculator will show you the value of that specific ratio.

Let's look at finding the tangent ratio by using a calculator.

\begin{align*}\text{tangent} \ 82^{\circ}\end{align*}

**You can find the values for each ratio using your calculator. When dealing with large decimals values, it is usually best to round the numbers to the nearest thousandth. It gives you a reasonably accurate value without being too long of a number to work with.**

**The tangent of \begin{align*}82^{\circ}\end{align*} is 7.11536972238419..., or about 7.115.**

Now we can use the tangent ratio to find the length of the height of a right triangle. We can see that the height in a right triangle looks like side \begin{align*}a\end{align*}. This is because of the type of triangle that a right triangle is. As you recall, the ratio of tangent is \begin{align*}\frac{opposite}{adjacent}\end{align*}. If you know the tangent value of the angle in question, and the length of the adjacent side, you can find the measure of the opposite side.

Look at the algebraic situation below.

\begin{align*}\text{tangent} \angle X&=\frac{opposite}{adjacent}\\ \text{tangent} \angle X \times adjacent &= \frac{opposite}{adjacent} \times adjacent\\ \text{tangent} \angle X \times adjacent &=opposite\end{align*}

**If you multiply the tangent of any angle \begin{align*}X\end{align*} and the length of the adjacent side, the result is the length of the opposite side.**

*Write this statement in your notebook. Be sure that it is connected to tangents.*

Take a look at this situation.

**What is the length of side \begin{align*}QR\end{align*} in the triangle below?**

**Use the following equation to find the length of the side opposite angle \begin{align*}P\end{align*}. Notice that to find the measure of the opposite side we need to first find the tangent of the angle we are working with. Then we can take that measure and multiply it with the adjacent side.**

\begin{align*}\text{tangent} \angle P \times adjacent &=opposite\\ \text{tangent} \ 72 \times 8 &=opposite\\ 3.078 \times 8 &= opposite\\ 24.624 &= opposite\end{align*}

**The length of side \begin{align*}QR\end{align*} is 24.624 units.**

Use a calculator to figure out each tangent. You may round to the nearest hundredth.

#### Example A

\begin{align*}\text{tangent} \ 32^{\circ}\end{align*}

**Solution: \begin{align*}0.62\end{align*}**

#### Example B

\begin{align*}\text{tangent} \ 15^{\circ}\end{align*}

**Solution: \begin{align*}0.27\end{align*}**

#### Example C

\begin{align*}\text{tangent} \ 89^{\circ}\end{align*}

**Solution: \begin{align*}57.29\end{align*}**

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

In this problem, you have a lot of information, but the only important data is the shape of the triangle, the length of the base, and the measurement of the angle. The base is the side adjacent to where Craig is standing, and you want to find the value of the side opposite where Craig was standing. For these purposes, the hypotenuse of the triangle is irrelevant.

Since you have the angle, and the adjacent side, you can find a tangent to discover the height of the triangle.

\begin{align*}\text{tangent} &=\frac{opposite}{adjacent}\\ \text{tangent} ~ 65^{\circ} &= \frac{opposite}{10 \ feet}\\ 2.145 &=\frac{opposite}{10 \ feet}\\ 2.145 \times 10 \ feet &= \frac{opposite}{10 \ feet} \times 10 \ feet\\ 21.45 \ feet &=opposite\end{align*}

**Craig’s rocket reached it’s highest height at 21.45 feet, almost 21 and \begin{align*}\frac{1}{2}\end{align*} feet high!**

### Vocabulary

- Sine
- a ratio between the opposite side and the hypotenuse of a given angle.

- Cosine
- a ratio between the adjacent side and the hypotenuse of a given angle.

- Tangent
- a ratio between the opposite side and the adjacent side of a given angle.

- Trigonometric Ratio
- used to find missing side lengths of right triangles when angle measures have been given.

### Guided Practice

Here is one for you to try on your own.

We have a right triangle. Angle A is 50 degrees. The side of the triangle adjacent to angle A is 4 inches. What is the length of the opposite side?

**Solution**

To figure this out, we can use the tangent of the angle and multiply times the length of the adjacent side.

Take a look.

\begin{align*}\text{tangent} \angle A \times adjacent &=opposite\\ \text{tangent} \ 50 \times 4 &=opposite\\ 1.19 \times 4 &= opposite\\ 4.76 &= opposite\end{align*}

**The length of the opposite side is 4.76 inches.**

### Video Review

Khan Academy Basic Trigonometry II

### Practice

Direction: Use a calculator to figure out each tangent. You may round to the nearest hundredth.

1. \begin{align*}\text{Tangent} \ 7^{\circ}\end{align*}

2. \begin{align*}\text{Tangent} \ 41^{\circ}\end{align*}

3. \begin{align*}\text{Tangent} \ 65^{\circ}\end{align*}

4. \begin{align*}\text{Tangent} \ 22^{\circ}\end{align*}

5. \begin{align*}\text{Tangent} \ 18^{\circ}\end{align*}

6. \begin{align*}\text{Tangent} \ 35^{\circ}\end{align*}

7. \begin{align*}\text{Tangent} \ 50^{\circ}\end{align*}

8. \begin{align*}\text{Tangent} \ 54^{\circ}\end{align*}

9. \begin{align*}\text{Tangent} \ 66^{\circ}\end{align*}

10. \begin{align*}\text{Tangent} \ 70^{\circ}\end{align*}

Direction: Find the length of the opposite side in each example.

11. We have a right triangle. Angle A is 45 degrees. The side of the triangle adjacent to angle A is 6 inches. What is the length of the opposite side?

12. We have a right triangle. Angle B is 63 degrees. The side of the triangle adjacent to angle B is 7 inches. What is the length of the opposite side?

13. We have a right triangle. Angle A is 29 degrees. The side of the triangle adjacent to angle A is 6 inches. What is the length of the opposite side?

14. We have a right triangle. Angle B is 12 degrees. The side of the triangle adjacent to angle B is 4.5 inches. What is the length of the opposite side?

15. We have a right triangle. Angle A is 9 degrees. The side of the triangle adjacent to angle A is 8 inches. What is the length of the opposite side?