# 5.4: Sine Ratio

**At Grade**Created by: CK-12

## Learning Objectives

- Review the different parts of right triangles.
- Identify and use the sine ratio in a right triangle.

## Review: Parts of a Triangle

The sine and cosine ratios relate **opposite** and **adjacent** sides to the **hypotenuse**. You already learned these terms in the previous lesson, but they are important to review and commit to memory.

The **hypotenuse** of a triangle is always *opposite the right angle* and is the longest side of a right triangle.

The terms **adjacent** and **opposite** depend on *which angle you are referencing:*

A side **adjacent** to an angle is the *leg of the triangle that helps form the angle.*

A side **opposite** to an angle is the *leg of the triangle that does not help form the angle.*

In your own words,

The **hypotenuse** is _____________________________________________________.

The **opposite** side is ____________________________________________________.

The **adjacent** side is ____________________________________________________.

**Example 1**

*Examine the triangle in the diagram below.*

*Identify which leg is adjacent to angle \begin{align*}N\end{align*}, which leg is opposite to angle \begin{align*}N\end{align*}, and which segment is the hypotenuse.*

The first part of the question asks you to identify the leg **adjacent** to \begin{align*}\angle{N}\end{align*}. Since an **adjacent** leg is the one that *helps to form the angle* and is not the hypotenuse, it must be \begin{align*}\overline{MN}\end{align*}.

The next part of the question asks you to identify the leg **opposite** \begin{align*}\angle{N}\end{align*}. Since an **opposite** leg is the leg that *does not help to form the angle,* it must be \begin{align*}\overline{LM}\end{align*}.

The **hypotenuse** is *always opposite the right angle,* so in this triangle it is segment \begin{align*}\overline{LN}\end{align*}.

**Reading Check:**

1. *Which side of a right triangle is the longest side?* _____________________________

2. *Describe where the side you answered in #1 above is in relation to the right angle:*

\begin{align*}\; \;\end{align*}

\begin{align*}\; \;\end{align*}

3. *Which side of a right triangle does not help to make the right angle?* _____________________________

4. *Which side of a right triangle helps to make the right angle and is NOT the hypotenuse?* _____________________________

## The Sine Ratio

Another important trigonometric ratio is **sine.** A sine ratio must always refer 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 to the length of the **hypotenuse.**

This means that the **sine** ratio is: the ____________________ side divided by the _______________________.

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*} \sin \theta = \frac{opposite}{hypotenuse}\end{align*}

**Example 2**

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

To find the solutions, you must identify the sides you need and build the ratios carefully. In the **sine** ratio, we will need the **opposite** side and the **hypotenuse.**

Remember, the **hypotenuse** of a right triangle is across from the right angle. The **opposite** side depends on which angle we are using.

The **hypotenuse** is the segment ___________, which is _______ cm long.

For angle \begin{align*}A\end{align*}:

The side **opposite** angle \begin{align*}A\end{align*} is the segment ___________, which is _______ cm long.

For angle \begin{align*}B\end{align*}:

The side **opposite** angle \begin{align*}B\end{align*} is the segment ___________, which is _______ cm long.

\begin{align*}\sin A & = \frac {opposite}{hypotenuse} = \frac {3}{5}\\ \sin B & = \frac {opposite}{hypotenuse} = \frac {4}{5}\end{align*}

So, \begin{align*}\sin A = \frac {3}{5}\end{align*} and \begin{align*}\sin B = \frac {4}{5}\end{align*}.

**Reading Check:**

*Your friend did the following problem and asked you if it was correct:*

** Find** \begin{align*}\sin X\end{align*}

*using the triangle below.*

\begin{align*}\sin X = \frac{opposite}{hypotenuse} = \frac{5}{12}\end{align*}

1. *Is your friend’s work correct? YES / NO (Circle your answer)*

\begin{align*}{\;}\end{align*}

\begin{align*}{\;} \end{align*}

2. *If not, where is the mistake in the problem? Describe the mistake in words and explain to your friend how she should have done the problem correctly.*

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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