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Warm Up

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Apply the Sine and Cosine Ratios

Warm Up

Lesson Presentation

Lesson Quiz

XZ

ANSWER

2.Name the leg oppositeX.

YZ

ANSWER

Warm-Up

Use this diagram for Exercises 1-4.

1.Name the hypotenuse.

3.Name the leg adjacent to X.

XY

ANSWER

4.IfXY = 17 andm X = 41 , findAC.

14.78

ANSWER

Warm-Up

Use this diagram for Exercises 1-4.

opp. S

RT

16

ST

63

=

=

=

0.9692

hyp

SR

SR

65

65

opp. R

=

=

=

0.2462

hyp

Example 1

Find sinSand sinR. Write each answer as a fraction and as a decimal rounded to four places.

SOLUTION

sinS

sinR

1.

ANSWER

8

or

0.4706,

17

15

or

0.8824

17

Guided Practice

Find sin Xand sin Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.

2.

ANSWER

3

or

0.6,

5

4

or

0.8

5

Guided Practice

Find sin Xand sin Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.

adj. to W

adj. to U

=

=

hyp

hyp

WV

18

24

UV

=

=

=

=

3

UW

30

UW

30

=

5

4

=

5

Example 2

Find cosU and cosW. Write each answer as a fraction and as a decimal.

SOLUTION

cosU

= 0.6000

cosW

= 0.8000

DOG RUN

You want to string cable to make a dog run from two corners of a building, as shown in the diagram. Write and solve a proportion using a trigonometric ratio to approximate the length of cable you will need.

Example 3

sin 35°

=

opp. hyp.

11

sin 35°

=

x

x sin 35°

= 11

x =

x

11

11

0.5736

sin35°

x

19.2

Example 3

SOLUTION

Write ratio for sine of 35o.

Substitute.

Multiply each side by x.

Divide each side by sin 35o.

Use a calculator to find sin 35o.

Simplify.

You will need a little more than 19 feet of cable.

3.

0.6, 0.8

ANSWER

Guided Practice

In Exercises 3 and 4, find cosRand cosS. Write each answer as a decimal. Round to four decimal places, if necessary.

4.

0.8824, 0.4706

ANSWER

Guided Practice

In Exercises 3 and 4, find cosRand cosS. Write each answer as a decimal. Round to four decimal places, if necessary.

about15.7 ft

ANSWER

5.

In Example 3, use the cosine ratio to find the length of the other leg of the triangle formed.

Guided Practice

SKIING

You are skiing on a mountain with an altitude of 1200 meters. The angle of depression is 21°. About how far do you ski down the mountain?

Example 4

sin 21°

=

opp. hyp.

sin 21°

1200

=

x

= 1200

x sin 21°

1200

x =

sin21°

1200

x

0.3584

x

3348.2

Example 4

SOLUTION

Write ratio for sine of 21o.

Substitute.

Multiply each side by x.

Divide each side by sin 21o

Use a calculator to find sin21o

Simplify.

You ski about 3348 meters down the mountain.

about2556 m

ANSWER

Guided Practice

6.WHAT IF? Suppose the angle of depression in Example 4 is 28°. About how far would you ski?

You want to build a skateboard ramp with a length of 14 feet and an angle of elevation of 26°. You need to find the height and length of the base of the ramp.

SKATEBOARD RAMP

Example 5

Find the height.

STEP 1

sin26°

=

x

=

opp. hyp.

sin26°

14

14 sin 26°

= x

6.1

x

Example 5

SOLUTION

Write ratio for sine of 26o.

Substitute.

Multiply each side by 14.

Use a calculator to simplify.

The height is about 6.1 feet.

cos26°

=

cos26°

y

=

adj. hyp.

14

14 cos 26°

= y

12.6

y

Example 5

Find the length of the base.

STEP 2

Write ratio for cosine of 26o.

Substitute.

Multiply each side by 14.

Use a calculator to simplify.

The length of the base is about 12.6 feet.

3

3

Use the 30° - 60° - 90° Triangle Theorem to draw a right triangle with side lengths of 1, , and 2. Then set up sine and cosine ratios for the 60° angle.

adj. hyp.

opp. hyp.

sin60°

=

0.08660

=

2

1

cos60°

=

=

0.5000

=

2

Example 6

Use a special right triangle to find the sine and cosine of a 60o angle.

SOLUTION

ANSWER

about 8 feet, about 11.5 ft

8.

Use a special right triangle to find the sine and cosine of a 30° angle.

3

,

ANSWER

2

1

2

Guided Practice

7.WHAT IF? In Example 5, suppose the angle of elevation is 35°. What is the new height and base length of the ramp?

Use this diagram for Exercises 1- 3.

1.

If x = , y = 4 andz = , find sinX, sinY, cosX and cosY.

4

4

ANSWER

cosY=

sinX=

0.9129,

0.9129

30

30

5

6

6

6

6

6

sinY=

cosX=

0.4082,

0.4082,

6

6

Lesson Quiz

Use this diagram for Exercises 1- 3.

2.

If y = 10 and m Y = 15°, find z to the nearest tenth.

ANSWER

38.6

Lesson Quiz

Use this diagram for Exercises 1- 3.

3.

If z = 12 and m X = 84°, find y to the nearest tenth.

ANSWER

1.3

Lesson Quiz

A lamppost is 11 feet tall. If the angle of elevation of the sun is 49° , how far is the top of the lamppost from the tip of its shadow?

4.

14.6 ft

ANSWER

Lesson Quiz