Have you ever built a ramp? Take a look at this dilemma.

Mr. Watson’s homeroom decided to do a community service repairing the ramp outside the shed. The fresh coat of paint shone in the bright sunlight and Mr. Watson walked across the grass with all of his students to look at the ramp outside the door of the shed.

“Has that always been there?” asked Dan.

“No, in fact it was just brought out here yesterday,” Mr. Watson explained.

“Well if it’s brand new, then why do we need to fix it?” Emily asked.

“Because it doesn’t fit neatly under the door.”

Sure enough, the students looked and could see that the back of the ramp was too tall and the students would need to fix the back to make it fit beneath the doorway. Fixing this ramp would help everyone because it would make it easy to push or pull the cart with the athletic equipment out onto the field and since the equipment was used by local kid’s teams as well as the school, this was a definite way to give back to the community.

“What do we need to do?” Dan asked.

They looked at the ramp. Mr. Watson drew the following sketch.

“Well that’s not very helpful,” Dan commented.

“Sure it is,” Emily said.

**Who is right? Is it possible to figure out the length of the missing side using Mr. Watson’s diagram? What kind of math will you need? This Concept will show you how to use trigonometric ratios to solve problems like this one.**

### 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.**

Let’s look at the first trigonometric ratio.

It is called the ** sine**.

**A sine refers to a particular angle in a right triangle. The sine of an angle is the ratio of the length of the leg opposite the angle we are focusing on to the length of the hypotenuse.**

Remember that in a ratio, you list the first item on top of the fraction and the second item on the bottom. So, the ratio of the sine will be \begin{align*}\frac{opposite}{hypotenuse}\end{align*}.

Let’s look at how we can find the sine of a particular angle.

**What are the sines of \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} in the triangle below?**

**To find the sine, all you have to do to is build the ratio carefully.**

\begin{align*}\text{sine} \angle A & = \frac{opposite}{hypotenuse} = \frac{3}{5} = 0.6\\ \text{sine} \angle B & = \frac{opposite}{hypotenuse} = \frac{4}{5} = 0.8\end{align*}

**Notice that once we have the ratio, that we can divide the numerator by the denominator to convert it to a decimal. The decimal is the answer that we are looking for with regard to the trigonometric ratio.**

**The sine of \begin{align*}\angle A\end{align*} is 0.6 and the sine of \begin{align*}\angle B\end{align*} is 0.8.**

**As you could see, the side opposite an angle is the one that an angle opens up to. An opposite side will never be one of the rays that forms and angle.**

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

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

#### Example B

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

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

#### Example C

What is the ratio for sine?

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

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

**In thinking about this problem, Emily is correct. You can use the diagram to solve the problem. To do this, you will need to use trigonometric ratios. In fact, you will need to use the sine ratio to figure out the height of the ramp.**

### 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.

### Guided Practice

Here is one for you to find on your own.

Find the sine of each acute angle in the diagram.

**Solution**

First, we need to identify the acute angles. We know that an acute angle is less than \begin{align*}90^\circ\end{align*}, so in this triangle, \begin{align*}\angle A\end{align*} and \begin{align*}\angle C\end{align*} are the acute angles. We will find the sine of each of these angles.

We know that the sine is the opposite over the hypotenuse. Here are the sine ratios.

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

### Video Review

Khan Academy Basic Trigonometry

### Practice

Directions: solve each problem.

- What is the sine of \begin{align*}\angle G\end{align*}?
- What is the sine of \begin{align*}\angle H\end{align*}?
- Can you find the sine of \begin{align*}\angle A\end{align*}?

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

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

Directions: Answer each question true or false.

- You can use the Pythagorean Theorem to find the length of a missing side in a right triangle.
- A right triangle must have a 90 degree angle.
- The sine ratio is the hypotenuse over the opposite side.
- If you know only side length, then you can figure out all the side lengths.
- Sine ratio has to do with side lengths.
- The hypotenuse is always opposite the right angle.
- You must be given all three side lengths to figure out the sine ratio.