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# Trigonometric Ratios with a Calculator

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

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Practice Trigonometric Ratios with a Calculator
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Trigonometric Ratios with a Calculator

What if you were given a 20-70-90 triangle? How could you find the sine, cosine, and tangent of the 20\begin{align*}20^\circ\end{align*} and 70\begin{align*}70^\circ\end{align*} angles? After completing this Concept, you'll be able to use a calculator to find the trigonometric ratios for angles that do not measure 45\begin{align*}45^\circ\end{align*}, 30\begin{align*}30^\circ\end{align*}, or 60\begin{align*}60^\circ\end{align*}.

### Watch This

CK-12 Foundation: The Trigonometric Ratios with a Calculator

James Sousa: Determining Trigonometric Function Values on the Calculator

### Guidance

There is a fixed sine, cosine, and tangent value for every angle, from 0\begin{align*}0^\circ\end{align*} to 90\begin{align*}90^\circ\end{align*}. Your scientific (or graphing) calculator knows all the trigonometric values for any angle. Your calculator, should have [SIN], [COS], and [TAN] buttons. You can use your calculator and the trigonometric ratios is to find the missing sides of a right triangle by setting up a trig equation.

#### Example A

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

a) sin78\begin{align*}\sin 78^\circ\end{align*}

b) cos60\begin{align*}\cos 60^\circ\end{align*}

c) tan15\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) sin78=0.97815\begin{align*}\sin 78^\circ = 0.97815\end{align*}

b) cos60=0.5\begin{align*}\cos 60^\circ = 0.5\end{align*}

c) tan15=0.26795\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 b\begin{align*}b\end{align*}, and cosine to find a\begin{align*}a\end{align*}. Use your calculator to evaluate the sine and cosine of the angles.

sin2230sin22b=b30=b11.2  cos22=a3030cos22=a   a27.8

#### Example C

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

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

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

CK-12 Foundation: The Trigonometric Ratios with a Calculator

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### Guided Practice

1. What is tan45\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 tan45=1\begin{align*}\tan 45^\circ=1\end{align*}?

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

tan2811tan28x=x11=x5.8  cos28=11y11cos28=y   y12.5

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

### Explore More

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.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.8.

### Vocabulary Language: English Spanish

trigonometry

trigonometry

The study of the relationships between the sides and angles of right triangles.
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.