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# Trigonometric Ratios with a Calculator

## Solving for values when triangles aren't special right triangles.

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Trigonometric Ratios with a Calculator

What if you wanted to find the missing sides of a right triangle with angles of \begin{align*}20^\circ\end{align*} and \begin{align*}70^\circ\end{align*} and a hypotenuse length of 10 inches? How could you use trigonometry to help you? After completing this Concept, you'll be able to solve problems like this one.

### Guidance

The trigonometric ratios are not dependent on the exact side lengths, but the angles. There is one fixed value for every angle, from \begin{align*}0^\circ\end{align*} to \begin{align*}90^\circ\end{align*}. Your scientific (or graphing) calculator knows the values of the sine, cosine and tangent of all of these angles. Depending on your calculator, you should have [SIN], [COS], and [TAN] buttons. Use these to find the sine, cosine, and tangent of any acute angle. One application of the trigonometric ratios is to use them to find the missing sides of a right triangle. All you need is one angle, other than the right angle, and one side.

#### Example A

Find the trigonometric value, using your calculator. Round to 4 decimal places.

a) \begin{align*}\sin 78^\circ\end{align*}

b) \begin{align*}\cos 60^\circ\end{align*}

c) \begin{align*}\tan 15^\circ\end{align*}

Depending on your calculator, you enter the degree and then press the trig button or the other way around. Also, make sure the mode of your calculator is in DEGREES.

a) \begin{align*}\sin 78^\circ = 0.97815\end{align*}

b) \begin{align*}\cos 60^\circ = 0.5\end{align*}

c) \begin{align*}\tan 15^\circ = 0.26795\end{align*}

#### Example B

Find the value of each variable. Round your answer to the nearest tenth.

We are given the hypotenuse. Use sine to find \begin{align*}b\end{align*}, and cosine to find \begin{align*}a\end{align*}. Use your calculator to evaluate the sine and cosine of the angles.

\begin{align*}\sin 22^\circ &= \frac{b}{30} && \quad \ \ \cos 22^\circ = \frac{a}{30}\\ 30 \cdot \sin 22^\circ &= b && 30 \cdot \cos 22^\circ = a\\ b & \approx 11.2 && \qquad \quad \ \ \ a \approx 27.8\end{align*}

#### Example C

Find the value of each variable. Round your answer to the nearest tenth.

We are given the adjacent leg to \begin{align*}42^\circ\end{align*}. To find \begin{align*}c\end{align*}, use cosine and use tangent to find \begin{align*}d\end{align*}.

\begin{align*}\cos 42^\circ &= \frac{adjacent}{hypotenuse} = \frac{9}{c} && \quad \tan 42^\circ = \frac{opposite}{adjacent} = \frac{d}{9}\\ c \cdot \cos 42^\circ &= 9 && 9 \cdot \tan 42^\circ = d\\ c &= \frac{9}{\cos 42^\circ} \approx 12.1 && \qquad \quad \ \ d \approx 27.0\end{align*}

Any time you use trigonometric ratios, use only the information that you are given. This will result in the most accurate answers.

Watch this video for help with the Examples above.

#### Concept Problem Revisited

Use trigonometric ratios to find the missing sides. Round to the nearest tenth.

Find the length of \begin{align*}a\end{align*} and \begin{align*}b\end{align*} using sine or cosine ratios:

\begin{align*}\cos 20^\circ = \frac{a}{10} && \sin 70^\circ = \frac{a}{10}\\ 10 \cdot \cos 20^\circ = a && 10 \cdot \sin 70^\circ = a\\ a \approx 9.4 && a \approx 9.4\\\\ \sin 20^\circ = \frac{b}{10} && \cos 70^\circ = \frac{b}{10}\\ 10 \cdot \sin 20^\circ = b && 10 \cdot \cos 70^\circ = b\\ b \approx 3.4 && b \approx 3.4\end{align*}

### Vocabulary

Trigonometry is the study of the relationships between the sides and angles of right triangles. The legs are called adjacent or opposite depending on which acute angle is being used. The three trigonometric (or trig) ratios are sine, cosine, and tangent.

### Guided Practice

1. What is \begin{align*}\tan 45^\circ\end{align*}?

2. Find the length of the missing sides and round your answers to the nearest tenth: .

3. Find the length of the missing sides and round your answers to the nearest tenth: .

1. Using your calculator, you should find that \begin{align*}\tan 45^\circ=1\end{align*}?

2. Use tangent for \begin{align*}x\end{align*} and cosine for \begin{align*}y\end{align*}.

\begin{align*}\tan 28^\circ &= \frac{x}{11} && \quad \ \ \cos 28^\circ = \frac{11}{y}\\ 11 \cdot \tan 28^\circ &= x && \frac{11}{\cos 28^\circ} = y\\ x & \approx 5.8 && \qquad \quad \ \ \ y \approx 12.5\end{align*}

3. Use tangent for \begin{align*}y\end{align*} and cosine for \begin{align*}x\end{align*}.

\begin{align*}\tan 40^\circ &= \frac{y}{16} && \quad \ \ \cos 40^\circ = \frac{16}{x}\\ 16 \cdot \tan 40^\circ &= y && \frac{16}{\cos 40^\circ} = x\\ y & \approx 13.4 && \qquad \quad \ \ \ x \approx 20.9\end{align*}

### Practice

Use your calculator to find the value of each trig function below. Round to four decimal places.

1. \begin{align*}\sin 24^\circ\end{align*}
2. \begin{align*}\cos 45^\circ\end{align*}
3. \begin{align*}\tan 88^\circ\end{align*}
4. \begin{align*}\sin 43^\circ\end{align*}
5. \begin{align*}\tan 12^\circ\end{align*}
6. \begin{align*}\cos 79^\circ\end{align*}
7. \begin{align*}\sin 82^\circ\end{align*}

Find the length of the missing sides. Round your answers to the nearest tenth.

1. Find \begin{align*}\sin 80^\circ\end{align*} and \begin{align*}\cos 10^\circ\end{align*}.
2. Use your knowledge of where the trigonometric ratios come from to explain your result to the previous question.
3. Generalize your result to the previous two questions. If \begin{align*}\sin\theta=x\end{align*}, then \begin{align*}\cos ?=x\end{align*}.
4. How are \begin{align*}\tan\theta \end{align*} and \begin{align*}\tan(90-\theta)\end{align*} related? Explain.

### Vocabulary Language: English

Hypotenuse

Hypotenuse

The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle

Legs of a Right Triangle

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Trigonometric Ratios

Trigonometric Ratios

Ratios that help us to understand the relationships between sides and angles of right triangles.