Apply the Sine and Cosine Ratios
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Apply the Sine and Cosine Ratios. Warm Up. Lesson Presentation. Lesson Quiz. XZ. ANSWER. 2. Name the leg opposite X. YZ. ANSWER. Warm-Up. Use this diagram for Exercises 1-4. 1. Name the hypotenuse. 3. Name the leg adjacent to X. XY. ANSWER.

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Warm Up

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Warm up

Apply the Sine and Cosine Ratios

Warm Up

Lesson Presentation

Lesson Quiz


Warm up

XZ

ANSWER

2.Name the leg oppositeX.

YZ

ANSWER

Warm-Up

Use this diagram for Exercises 1-4.

1.Name the hypotenuse.


Warm up

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.


Warm up

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


Warm up

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.


Warm up

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.


Warm up

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


Warm up

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


Warm up

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.


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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.


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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.


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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


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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


Warm up

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.


Warm up

about2556 m

ANSWER

Guided Practice

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


Warm up

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


Warm up

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.


Warm up

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.


Warm up

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


Warm up

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?


Warm up

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


Warm up

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


Warm up

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


Warm up

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


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