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Vocabulary

Review

HSM11_A1MC_1006_T91381

HSM11_A1MC_1006_T91382HSM11_A1MC_1006_T91383

Chapter 10 310

10-6 Trigonometric Ratios

Use the circles, triangles, and squares below. Write each ratio.

1. number of circles to number of squares 5

2. number of squares to number of circles 5

3. total number of shapes to number of triangles 5

Vocabulary Builder

elevation (noun) el uh vay shun

Definition: An object’s elevation is its height above a surface or other object.

Related Words: elevate (verb), elevator (noun), horizontal (adjective, noun)

Opposite: depression (noun)

Main Idea: The angle of elevation is the angle of a line of sight above the horizontal. The angle of depression is the angle of a line of sight below the horizontal.

Use Your Vocabulary

Write elevation or depression to indicate the type of angle that is shown.

4. 5.

angle of

angle of

5

4

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Problem 1

Key Concept Trigonometric Ratios

HSM11_A1MC_1006_T91384

Name Written Definition

sine of /A sin A

cosine of /A cos A

tangent of /A tan A

opposite leghypotenuse

adjacent leghypotenuse

opposite legadjacent leg

hypotenuse

leg adjacent to A

leg opposite A

A

hsm11a1se_1006_t02132C

HSM11_A1MC_1006_T91385

C

HSM11_A1MC_1006_T91386

F

G E

9

12

15

Normal S c i EngF loat 0123 45678 9Rad ian Deg reeFunc Pa r Po l SeqConnec ted DotSequent ia l S imu lRea l a+b i r e^ 0 iFu l l Ho r iz G -T

Set your calculatorto Degree mode.

311 Lesson 10-6

Finding Trigonometric Ratios

Got It? What are sin E, cos E, and tan E for the triangle at the right?

8. Underline the correct words to complete the sentence.

For angle E, the opposite leg is side GE / side FG ,

and the adjacent leg is side GE / side EF .

9. Complete each equation.

sin E 5opposite leghypotenuse cos E 5

adjacent leghypotenuse tan E 5

opposite legadjacent leg

515

55

515

55

5 5

You can also use a calculator to find trigonometric ratios. In this chapter, use the degree mode when finding trigonometric ratios. That allows you to enter angles in degrees.

For each triangle, label the hypotenuse, opposite leg, and adjacent leg for angle C.

6. 7.

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Problem 4

Problem 3

HSM11_A1MC_1006_T91387

A

B C

1.57535

x

hsm11a1se_1006_t02138

Chapter 10 312

Finding a Missing Side Length

Got It? To the nearest tenth, what is the value of x in the triangle at the right?

10. Circle the trigonometric equation that can be used to solve for x.

sin 358 5opposite leghypotenuse cos 358 5

adjacent leghypotenuse tan 358 5

opposite legadjacent leg

11. Complete the steps below to solve the equation.

5x

Substitute x and 1.575 from the diagram.

cos 358 ? x 5 Multiply each side by x.

x 5 Divide each side of the equation by cos 358.

x < Use a calculator to find cos 358. Then divide and round to the nearest tenth.

If you know the lengths of two sides of a right triangle, you can find a trigonometric ratio for each acute angle of the triangle. If you know a trigonometric ratio for an angle, you can use the inverse of that ratio to find the measure of the angle. Use the inverse

trigonometric features sin21 , cos21 , and tan21 on your calculator.

Finding the Measures of Angles

Got It? In a right triangle, the side opposite lA is 8 mm long and the hypotenuse is 12 mm long. What is the measure of lA?

12. Label the diagram at the right to show which side is 8 mm long and which side is 12 mm long.

13. Circle the trigonometric function you can use to find the measure of angle A if you know the lengths of the opposite side and the hypotenuse.

sin A cos A tan A

14. Circle the correct meaning for the trigonometric function sin21x.

the angle whose sine is x the sine of x

15. Complete the equation and solve.

5 8 Use a trigonometric definition.

sin A < Divide.

A < Use the sin21 function of a calculator.

16. To the nearest degree, angle A measures .

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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

Problem 5

HSM11_A1MC_1006_T91388

150 ft

50°xeye level

top of ride

313 Lesson 10-6

• Do you UNDERSTAND?

Vocabulary Describe the difference between finding the sine of an angle and the cosine of an angle.

19. Circle the ratio that is used to find the sine of an angle. Draw a line under the ratio that is used to find the cosine of an angle.

adjacent leghypotenuse

opposite legadjacent leg

opposite leghypotenuse

20. Describe how finding the sine of an angle differs from finding the cosine.

______________________________________________________________________________________

______________________________________________________________________________________

Check off the vocabulary words that you understand.

trigonometric ratios angle of elevation angle of depression

Rate how well you can work with trigonometric ratios.

Using an Angle of Elevation or Depression

Got It? Suppose that you are waiting in line for an amusement park ride. You see your friend at the top of the ride. The angle of elevation to the top of the ride is 508. How far are you from the base of the ride?

17. Complete the steps to find the value of x.

5x

Use the definition of tangent.

? x 5 150 Multiply each side by x.

x 5 150 Solve for x.

x < Use a calculator.

18. To the nearest foot, you are ft from the base of the ride.

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