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?

#### Guidance

Two right triangles with one pair of non-right congruent angles are similar by \begin{align*}AA \sim\end{align*}

For example, in the picture above, \begin{align*}\frac{a}{c}=\frac{d}{f}\end{align*}**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*}opposite legadjacent leg .*The abbreviation for tangent is tan.* - The
**sine**of an angle gives the ratio \begin{align*}\frac{\text{opposite leg}}{\text{hypotenuse}}\end{align*}opposite leghypotenuse .*The abbreviation for sine is sin.* - The
**cosine**of an angle gives the ratio \begin{align*}\frac{\text{adjacent leg}}{\text{hypotenuse}}\end{align*}adjacent leghypotenuse .*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.

**Example A**

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

**Solution:** \begin{align*}\sin (27^\circ) \approx 0.454\end{align*}

**Example B**

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

**Solution:** Look to see how the sides that are marked are related to the \begin{align*}27^\circ\end{align*}**adjacent** to the angle. The side of length \begin{align*}x\end{align*}**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*}

**Example C**

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

**Solution:** Relative to angle \begin{align*}\theta\end{align*}

\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*}

**Concept Problem Revisited**

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.

#### Vocabulary

Two figures are ** similar** if a similarity transformation will carry one figure to the other.

**will always have corresponding angles congruent and corresponding sides proportional.**

*Similar figures*
** AA, or Angle-Angle**, is a criterion for triangle similarity. The AA criterion for triangle similarity states that if two triangles have two pairs of congruent angles, then the triangles are similar.

The ** tangent (tan)** of an angle within a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

The ** sine (sin)** of an angle within a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

The ** cosine (cos)** of an angle within a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

The ** trigonometric ratios** are sine, cosine, and tangent.

** Trigonometry** is the study of triangles.

\begin{align*}\theta\end{align*}** “theta”,** is a Greek letter. In geometry, it is often used as a variable to represent an unknown angle measure.

#### Guided Practice

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

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

3. Find \begin{align*}\sin \theta\end{align*}

**Answers:**

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

2. Look to see how the sides that are marked are related to the \begin{align*}39^\circ\end{align*}**opposite** from the angle. The side of length \begin{align*}x\end{align*}**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*}

3. Relative to angle \begin{align*}\theta\end{align*}

\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.

#### Practice

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?