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Trigonometric Ratios with a Calculator

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

Guidance

There is a fixed sine, cosine, and tangent value for every angle, from $0^\circ$ to $90^\circ$ . 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) $\sin 78^\circ$

b) $\cos 60^\circ$

c) $\tan 15^\circ$

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) $\sin 78^\circ = 0.97815$

b) $\cos 60^\circ = 0.5$

c) $\tan 15^\circ = 0.26795$

Example B

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

We are given the hypotenuse. Use sine to find $b$ , and cosine to find $a$ . Use your calculator to evaluate the sine and cosine of the angles.

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

Example C

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

We are given the adjacent leg to $42^\circ$ . To find $c$ , use cosine and use tangent to find $d$ .

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

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

Guided Practice

1. What is $\tan 45^\circ$ ?

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 $\tan 45^\circ=1$ ?

2. Use tangent for $x$ and cosine for $y$ .

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

3. Use tangent for $y$ and cosine for $x$ .

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

Practice

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

1. $\sin 24^\circ$
2. $\cos 45^\circ$
3. $\tan 88^\circ$
4. $\sin 43^\circ$
5. $\tan 12^\circ$
6. $\cos 79^\circ$
7. $\sin 82^\circ$

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