SOH-CAH-TOA is a mnemonic that many people use to remember the difference between sine, cosine, and tangent. How can remembering SOH-CAH-TOA help you?

### Sine and Cosine Ratios

Two right triangles with one pair of non-right congruent angles are similar by \begin{align*}AA \sim\end{align*}. This means the ratio between the side lengths of the first triangle must be congruent to the ratio between the corresponding side lengths of the second triangle.

For example, in the picture above, \begin{align*}\frac{a}{c}=\frac{d}{f}\end{align*}. Because there are **three** ** pairs** of sides for any triangle, there are

**three**relevant

**ratios**for a given angle.

- The
**tangent**of an angle gives the ratio \begin{align*}\frac{\text{opposite leg}}{\text{adjacent leg}}\end{align*}.*The abbreviation for tangent is tan.* - The
**sine**of an angle gives the ratio \begin{align*}\frac{\text{opposite leg}}{\text{hypotenuse}}\end{align*}.*The abbreviation for sine is sin.* - The
**cosine**of an angle gives the ratio \begin{align*}\frac{\text{adjacent leg}}{\text{hypotenuse}}\end{align*}.*The abbreviation for cosine is cos.*

These are the basic **trigonometric** **ratios**. **Trigonometry** is the study of triangles. These ratios are called trigonometric ratios because they apply to triangles. Just as your scientific or graphing calculator has **tangent** programmed into it, it also has **sine** and **cosine** programmed into it. This means that you can use your calculator to determine the ratio between the lengths of any pair of sides for any angle within a right triangle.

**Solve the following problem**

Use your calculator to find the sine ratio and cosine ratio for a \begin{align*}27^\circ\end{align*} angle.

\begin{align*}\sin (27^\circ) \approx 0.454\end{align*} and \begin{align*}\cos (27^\circ) \approx 0.891\end{align*}.

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

Look to see how the sides that are marked are related to the \begin{align*}27^\circ\end{align*} angle. The side of length 11 is **adjacent** to the angle. The side of length \begin{align*}x\end{align*} is the **hypotenuse** of the triangle. When working with the adjacent side and the hypotenuse, you should use the cosine ratio.

\begin{align*}\cos (27^\circ) &= \frac{\text{adjacent leg}}{\text{hypothenuse}}\\ \cos (27^\circ) &= \frac{11}{x}\\ x &= \frac{11}{\cos (27^\circ)}\\ x & \approx \frac{11}{0.891}\\ x & \approx 12.346\end{align*}

Note that \begin{align*}\frac{11}{\cos (27^\circ)}\end{align*} is the exact answer. 12.346 is an approximate answer because you rounded the value of \begin{align*}\cos (27^\circ)\end{align*}.

#### Find \begin{align*}\sin \theta\end{align*} and \begin{align*}\cos \theta\end{align*}.

Relative to angle \begin{align*}\theta\end{align*}, 3 is the opposite leg and 4 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use the Pythagorean Theorem to find the hypotenuse:

\begin{align*}3^2+4^2 &= h^2\\ 25 &= h^2\\ h &= 5\end{align*}

Now, write the sine and cosine ratios:

\begin{align*}\sin \theta &= \frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{3}{5}\\ \cos \theta &= \frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{4}{5}\end{align*}

**Examples**

**Example 1**

Earlier, you were asked how can remembering SOH-CAH-TOA help you.

SOH-CAH-TOA is a mnemonic that many people use to remember the difference between sine, cosine, and tangent. How can remembering SOH-CAH-TOA help you?

SOH-CAH-TOA stands for **S**ine equals **O**pposite over **H**ypotenuse, **C**osine equals **A**djacent over **H**ypotenuse, and **T**angent equals **O**pposite over **A**djacent. This mnemonic helps you remember which trigonometric ratio is which.

#### Example 2

Use your calculator to find the sine and cosine ratios for a \begin{align*}39^\circ\end{align*} angle.

\begin{align*}\sin (39^\circ) \approx 0.629\end{align*} and \begin{align*}\cos (39^\circ) \approx 0.777\end{align*}.

#### Example 3

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

Look to see how the sides that are marked are related to the \begin{align*}39^\circ\end{align*} angle. The side of length 17 is **opposite** from the angle. The side of length \begin{align*}x\end{align*} is the **hypotenuse** of the triangle. When working with the opposite side and the hypotenuse, you should use the sine ratio.

\begin{align*}\sin (39^\circ) &= \frac{\text{opposite leg}}{\text{hypotenuse}}\\ \sin (39^\circ) &= \frac{17}{x}\\ x &= \frac{17}{\sin (39^\circ)}\\ x & \approx \frac{11}{0.629}\\ x & \approx 27.013\end{align*}

#### Example 4

Find \begin{align*}\sin \theta\end{align*} and \begin{align*}\cos \theta\end{align*}.

Relative to angle \begin{align*}\theta\end{align*}, 4 is the opposite leg and 2 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use the Pythagorean Theorem to find the hypotenuse:

\begin{align*}4^2+2^2 &= h^2\\ 20 &= h^2\\ h &= 2 \sqrt{5} \end{align*}

Now, write the sine and cosine ratios:

\begin{align*}\sin \theta &= \frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{4}{2 \sqrt{5}}=\frac{2}{\sqrt{5}}=\frac{2 \sqrt{5}}{5}\\ \cos \theta &= \frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{2}{2 \sqrt{5}}=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\end{align*}

Note that in the last step of each calculation the denominator was rationalized. You can choose to rationalize the denominator if you wish.

### Review

For #1-#6, use the triangle below. Find each exact value.

1. \begin{align*}\sin E\end{align*}

2. \begin{align*}\cos E\end{align*}

3. \begin{align*}\tan E\end{align*}

4. \begin{align*}\sin F\end{align*}

5. \begin{align*}\cos F\end{align*}

6. \begin{align*}\tan F\end{align*}

Identify whether the sine, cosine, or tangent ratio is most useful for helping to solve the problem. Then, solve for \begin{align*}x\end{align*}.

7.

8.

9.

10.

11.

12.

Use the triangle below for #13-#15.

13. Find \begin{align*}m \angle C\end{align*}.

14. Draw an altitude from \begin{align*}\angle B\end{align*} to divide the triangle into two right triangles. Use trigonometry to find the lengths of the sides of each of these right triangles.

15. Find the perimeter of \begin{align*}\Delta ABC\end{align*}.

16. True or false: You need to know the length of at least one side of a triangle to find the lengths of the other sides.

17. What are the trigonometric ratios?

18. What do the trigonometric ratios have to do with similar triangles?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.2.