CorrectionKey=NL-B;CA-B CorrectionKey=NL-D;CA-D 18 . 2 DO … · 2019. 2. 21. · ab = __ = _...

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Explore Investigating Ratios in a Right TriangleYou can use geometry software or an online tool to explore ratios of side lengths in right triangles.

Construct three points A, B, and C.Construct rays

‾→ AB and

‾→ AC . Move C so that ∠A is acute.

Construct point D on _ AC . Construct a line through D

perpendicular to _ AB . Construct point E as the intersection of

the perpendicular line and _ AB .

Measure ∠A. Measure the side lengths DE, AE, and AD of △ADE.

Calculate the ratios DE _ AD and AE _ AD .

Reflect

1. Drag D along ‾→ AC . What happens to m∠A as D moves along

‾→ AC ? What postulate or

theorem guarantees that the different triangles formed are similar to each other?

2. As you move D along ‾→ AC , what happens to the values of the ratios DE _ AD and AE _ AD ?

Use the properties of similar triangles to explain this result.

3. Move C. What happens to m∠A? With a new value of m∠A, note the values of the two ratios. What happens to the ratios if you drag D along

‾→ AC ?

Resource Locker

m∠A does not change; AA Similarity Theorem, given also that ∠AED remains a right angle.

The ratios do not change; Given similar triangles △ADE, and △AD’E’,

AD _ AD’

= DE _ D’E’

→ D’E’ _ AD’

= DE _ AD

and AD _ AD’

= AE _ AE’

→ AE’ _ AD’

= AE _ AD

.

m∠A changes in value; the new values of the ratios do not change as D is dragged

along ‾→ AC .

Module 18 941 Lesson 2

18 . 2 Sine and Cosine RatiosEssential Question: How can you use the sine and cosine ratios, and their inverses,

in calculations involving right triangles?

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Common Core Math StandardsThe student is expected to:

G-SRT.C.6

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Also G-SRT.C.7, G-SRT.C.8

Mathematical Practices

MP.4 Modeling

Language ObjectiveExplain to a partner how to find the sine and cosine of an angle given a diagram of a right triangle with given angle measure and opposite or adjacent leg and hypotenuse lengths.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 725734

Turn to these pages to find this lesson in the hardcover student edition.

Sine and Cosine Ratios

ENGAGE Essential Question: How can you use the sine and cosine ratios, and their inverses, in calculations involving right triangles?Given the measure of one acute angle and the

length of the hypotenuse, you can use the sine and

cosine ratios to find the lengths of each leg. Given

the length of the hypotenuse and the length of a

leg, you can use their inverses to find the measure

of each acute angle.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photograph, asking students to speculate on what the person in the photo might be doing. Then preview the Lesson Performance Task.

941

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore Investigating Ratios in a Right Triangle

You can use geometry software or an online tool to explore ratios of side lengths in right triangles.

Construct three points A, B, and C.

Construct rays ‾→ AB and

‾→ AC . Move C so that ∠A is acute.

Construct point D on _ AC . Construct a line through D

perpendicular to _ AB . Construct point E as the intersection of

the perpendicular line and _ AB .

Measure ∠A. Measure the side lengths DE, AE, and

AD of △ADE.

Calculate the ratios DE _ AD

and AE _ AD

.

Reflect

1. Drag D along ‾→ AC . What happens to m∠A as D moves along

‾→ AC ? What postulate or

theorem guarantees that the different triangles formed are similar to each other?

2. As you move D along ‾→ AC , what happens to the values of the ratios DE _

AD and AE _

AD ?

Use the properties of similar triangles to explain this result.

3. Move C. What happens to m∠A? With a new value of m∠A, note the values of

the two ratios. What happens to the ratios if you drag D along ‾→ AC ?

Resource

Locker

G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in

the triangle, leading to definitions of trigonometric ratios for acute angles. Also G-SRT.C.7,

G-SRT.C.8

COMMONCORE

m∠A does not change; AA Similarity Theorem, given also that ∠AED remains a right angle.

The ratios do not change; Given similar triangles △ADE, and △AD’E’,

AD _

AD’ =

DE _ D’E’

→ D’E’ _ AD’

= DE _

AD and A

D _ AD’

= AE _

AE’ → A

E’ _ AD’

= AE _

AD .

m∠A changes in value; the new values of the ratios do not change as D is dragged

along ‾→ AC .

Module 18

941

Lesson 2

18 . 2 Sine and Cosine Ratios

Essential Question: How can you use the sine and cosine ratios, and their inverses,

in calculations involving right triangles?

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941 Lesson 18 . 2

L E S S O N 18 . 2

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Explain 1 Finding the Sine and Cosine of an Angle

Trigonometric Ratios

A trigonometric ratio is a ratio of two sides of a right triangle. You have already seen one trigonometric ratio, the tangent. There are two additional trigonometric ratios, the sine and the cosine, that involve the hypotenuse of a right triangle.

The sine of ∠A, written sin A, is defined as follows:

sin A = length of leg opposite ∠A

___ length of hypotenuse

= BC _ AB

The cosine of ∠A, written cos A, is defined as follows:

cos A = length of leg adjacent to ∠A


= AC _ AB

You can use these definitions to calculate trigonometric ratios.

Example 1 Write sine and cosine of each angle as a fraction and as a decimal rounded to the nearest thousandth.

∠D

sin D = length of leg opposite ∠D


= EF _ DF = 8 _ 17 ≈ 0.471

cos D = length of leg adjacent to ∠D


= DE _ DF = 15 _ 17 ≈ 0.882

∠F

sin F = length of leg opposite to ∠F


= DE _ DF = _ ≈

cos F = length of leg adjacent to ∠F


= _ ≈

Reflect

4. What do you notice about the sines and cosines you found? Do you think this relationship will be true for any pair of acute angles in a right triangle? Explain.

5. In a right triangle △PQR with PR = 5, QR = 3, and m∠Q = 90°, what are the values of sin P and cos P?

sin P =

cos P =

A

B

C

E

15

8

17

F

D

15

170.882

0.4718

3 _ 5

4 _ 5

17

sin D = cos F and cos D = sin F; this relationship always holds because the leg opposite one

acute angle is adjacent to the other one.


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EXPLORE Investigating Ratios in a Right Triangle

INTEGRATE TECHNOLOGYStudents have the option of doing the Explore activity either in the book or online.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Have students examine what happens to the ratio of the opposite side length to the hypotenuse length as the acute angle gets closer to 90°. Repeat for the ratio of the adjacent side length to the hypotenuse length as the acute angle gets closer to 0°.

QUESTIONING STRATEGIESIf the measure of the acute angle of the right triangle does not change but the side lengths

of the triangle change, how do the ratios change? Explain. The values of the numerator and the

denominators will change but the ratios are equal

to the original ratios of opposite length to

hypotenuse and adjacent length to hypotenuse

because the triangles are similar.

EXPLAIN 1 Finding the Sine and Cosine of an Angle

AVOID COMMON ERRORSStudents often use the wrong ratio for sine or cosine. Help students review these relationships by using flashcards, mnemonics, or other memory aids. Encourage students to research mnemonics or produce their own.

PROFESSIONAL DEVELOPMENT

Math BackgroundTrigonometry is the branch of mathematics concerned with angle relationships in triangles. The ancient Egyptians used trigonometry to reset land boundaries after the Nile River flooded each year. The Babylonians used trigonometry to measure distances to nearby stars. Trigonometry is used in modern engineering, cartography, medical imaging, and many other fields.

Sine and Cosine Ratios 942


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Explain 2 Using Complementary AnglesThe acute angles of a right triangle are complementary. Their trigonometric ratios are related to each other as shown in the following relationship.

Trigonometric Ratios of Complementary Angles

If ∠A and ∠B are the acute angles in a right triangle, then sin A = cos B and cos A = sin B.

Therefore, if θ (“theta”) is the measure of an acute angle, then sin θ = cos (90° - θ) and cos θ = sin (90° - θ) .

You can use these relationships to write equivalent expressions.

Example 2 Write each trigonometric expression.

Given that sin 38° ≈ 0.616, write the cosine of a complementary angle in terms of the sine of 38°. Then find the cosine of the complementary angle.

Use an expression relating trigonometric ratios of complementary angles.

sin θ = cos (90° - θ)

Substitute 38 into both sides. sin 38° = cos (90° - 38°)

Simplify. sin 38° = cos 52°

Substitute for sin 38°. 0.616 ≈ cos 52°

So, the cosine of the complementary angle is about 0.616.

Given that cos 60° = 0.5, write the sine of a complementary angle in terms of the cosine of 60°. Then find the sine of the complementary angle.

Use an expression relating trigonometric ratios of complementary angles.

cos θ = sin (90° - θ)

Substitute into both sides. cos ° = sin (90° - °) Simplify the right side. cos ° = sin °

Substitute for the cosine of °. = sin °

So, the sine of the complementary angle is 0.5.

A

B

Cθ

(90° - θ)

60 60

60

300.5

60

60

30




COLLABORATIVE LEARNING

Small Group ActivityHave students work in small groups to investigate the relationship between the size of an angle and its sine, using geometry software or graph paper, rulers, and protractors. They should draw several triangles with unit side lengths and an angle that increases. Those using graph paper will need to measure with protractor and ruler. Have them determine how the sine changes as the angle size increases; then repeat for cosine. Have students share their results and how they drew their conclusions.

QUESTIONING STRATEGIESIf you know only the sine for an acute angle of a right triangle, how could you find the

cosine? The sine gives the ratio of the opposite side

to the hypotenuse, so you can construct a right

triangle with a hypotenuse and a leg that match the

ratio. Then you can use the Pythagorean Theorem to

find the length of the other leg, and use it to write

the cosine ratio.

Do you think it is possible for the value of a sine or cosine to be greater than 1? Why or

why not? It is not possible; because the hypotenuse

is the longest side of a right triangle, any ratio that

has the length of the hypotenuse as the

denominator will be less than 1.

EXPLAIN 2 Using Complementary Angles

QUESTIONING STRATEGIESWhy do the sine and cosine have a complementary angle relationship? The ratios

are for a right triangle. Since one angle must be a

right angle, the other two angles must be

complementary because the sum of the measures of

the angles of a triangle is 180°.

How can you write equivalent expressions for sin x° using cosine and cos y° using sine?

Explain. Use the complement to write

sin x° = cos (90° - x°) and cos y° = sin (90° - y°) .How can the relationship between the sine and cosine of complementary angles help you

solve equations involving sines and cosines? Apply

the fact that the total measure of the angles is 90° to

set up an equation to solve for unknown values.

943 Lesson 18 . 2


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Reflect

6. What can you conclude about the sine and cosine of 45°? Explain.

7. Discussion Is it possible for the sine or cosine of an acute angle to equal 1? Explain.

Your Turn

Write each trigonometric expression.

8. Given that cos 73° ≈ 0.292, write the sine of a complementary angle.

9. Given that sin 45° ≈ 0.707, write the cosine of a complementary angle.

Explain 3 Finding Side Lengths using Sine and CosineYou can use sine and cosine to solve real-world problems.

Example 3 A 12-ft ramp is installed alongside some steps to provide wheelchair access to a library. The ramp makes an angle of 11° with the ground. Find each dimension, to the nearest tenth of a foot.

Find the height x of the wall.

Use the definition of sine. sin A = length of leg opposite ∠A


= AB _ AC

Substitute 11° for A, x for BC, and 12 for AC. sin 11° = x _ 12

Multiply both sides by 12. 12sin 11° = x

Use a calculator to evaluate the expression. x ≈ 2.3

So, the height of the wall is about 2.3 feet.

B

C

wall

y

x 12 ft11°

A

θ = 73°, so 90 - θ = 17°.

sin 17° ≈ 0.292

θ = 45°, so 90 - θ = 45°.

cos 45° ≈ 0.707

No; the hypotenuse of a right triangle is always longer than its legs, so the side ratios

defining sine and cosine must always be less than 1.

sin 45° = cos 45°; 45° is complementary to itself.





DIFFERENTIATE INSTRUCTION

Critical ThinkingDiscuss how to use the trigonometric ratios to prove that tan x = sinx ____ cosx . Ask students if they find this surprising and why or why not.

COLLABORATIVE LEARNINGHave students construct right triangles with unit side lengths on graph paper, then use a protractor to measure each of the acute angles to the nearest degree. Have students verify the complementary relationship between the sine and cosine for their triangles for each of the acute angles, and then share their results with the class.

EXPLAIN 3 Finding Side Lengths using Sine and Cosine

INTEGRATE TECHNOLOGYIt may be helpful to review with students how to evaluate expressions of the form asin (x°) and bcos (y°) , with given values for a, b, x, and y, using their calculators.

QUESTIONING STRATEGIESWhy is a trigonometric ratio useful in solving a real-world problem involving right

triangles? Sample answer: It makes it possible to

use known lengths and angle measures to find

unknown lengths that might be difficult to measure.

If you use a trigonometric ratio, such as sine or cosine, to find the length of one of the legs

of a right triangle, do you have to use a trigonometric ratio to find the length of the other leg as well? Explain. No, you could also use the Pythagorean

Theorem to find the other leg since you will know

the lengths of the hypotenuse and one leg.



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B Find the distance y that the ramp extends in front of the wall.

Use the definition of cosine. cos A = length of leg adjacent to ∠A


= AB _ AC

Substitute ° for A, y for AB, and for AC. cos ° = y _

Multiply both sides by . cos ° = y

Use a calculator to evaluate the expression. y ≈

So, the ramp extends in front of the wall about feet.

Reflect

10. Could you find the height of the wall using the cosine? Explain.

Your Turn

11. Suppose a new regulation states that the maximum angle of a ramp for wheelchairs is 8°. At least how long must the new ramp be? Round to the nearest tenth of a foot.

Explain 4 Finding Angle Measures using Sine and CosineIn the triangle, sinA = 5 _ 10 = 1 _ 2 . However, you already know that sin 30° = 1 _ 2 . So you can

conclude that m∠A = 30°,

and write sin -1 ( 1 _ 2 ) = 30°.

Extending this idea, the inverse trigonometric ratios for sine and cosine are defined as follows:

Given an acute angle, ∠A,

• if sin A = x, then sin -1 x = m∠A, read as “inverse sine of x” • if cos A = x, then cos -1 x = m∠A, read as “inverse cosine of x”

You can use a calculator to evaluate inverse trigonometric expressions.

A

105

C

B

B

C

wallz

2.3 ft

8°A

11.8

11.8

Since ∠A and ∠B are complementary, m∠C = 79°. Then cos 79° = x _ 12

and

x = 12cos 79° ≈ 2.3 ft.

111211

12

12

1112

The ramp must be at least long enough to create an 8° angle at ∠A.

sin A = BC _ AC

⇒ sin 8° ≈ 2.3 _ z

z ≈ 16.5

The ramp must be at least 16.5 ft long.


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LANGUAGE SUPPORT

Connect Vocabulary Distinguishing between sine and cosine may be challenging for some students. Explain that the prefix co- can mean together, as it does in the word cooperate. Point out that the cosine ratio for an acute angle of a triangle involves the adjacent leg. Tell students to remember this by thinking of the adjacent leg as “coming together” with the hypotenuse from the angle.

EXPLAIN 4 Finding Angle Measures using Sine and Cosine

QUESTIONING STRATEGIESIn the equation y = si n -1 x, explain what x and y represent. In the equation, x represents

the sine of the angle with measure y°.

945 Lesson 18 . 2


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Example 4 Find the acute angle measures in △PQR, to the nearest degree.

Write a trigonometric ratio for ∠R.

Since the lengths of the hypotenuse and the opposite leg are given,

use the sine ratio. sinR = PQ

_ PR

Substitute 7 for PQ and 13 for PR. sinR = 7 _ 13

Write and evaluate an inverse trigonometric ratio to find m∠R and m∠P.

Start with the trigonometric ratio for ∠R. sinR =

Use the definition of the inverse sine ratio. m∠R = sin -1

Use a calculator to evaluate the inverse sine ratio. m∠R = °

Write a cosine ratio for ∠P. cosP = PQ

_ PR

Substitute for PQ and for PR. cosP =

Use the definition of the inverse cosine ratio. m∠P = co s - 1

Use a calculator to evaluate the inverse cosine ratio. m∠P = °

Reflect

12. How else could you have determined m∠P?

Your Turn

Find the acute angle measures in △XYZ, to the nearest degree.

13. m∠Y

14. m∠Z

Elaborate

15. How are the sine and cosine ratios for an acute angle of a right triangle defined?

Q 7

13

R

P

Z

23 14

Y

X

A

hypotenuse

adjacent

opposite

C

B

7 __ 13

7 __ 13

7 __ 13

7 __ 13

33

7 13

57

∠P and ∠R are complementary. Therefore, m∠P = 90° - m∠R ≈ 90° - 33° = 57°.

cos Y = XY _ YZ

= 14 _ 23

m∠Y = cos -1 ( 14 _ 23

) ≈ 53°

m∠Z = 90° - m∠Y ≈ 90° - 53° = 37°

sin A = opposite

__ hypotenuse

= BC _ AB

and cos A = adjacent

__ hypotenuse

= AC _ AB





QUESTIONING STRATEGIESHow could you evaluate the inverse sine or cosine of an angle without using the

calculator’s inverse trigonometric keys or a table of values? Use the measurements of the sine or cosine

ratio in the Pythagorean Theorem to find the length

of the missing leg. Then draw a right triangle with

the side lengths and use a protractor to measure the

angle formed by the hypotenuse and the opposite

side or the adjacent side to estimate the angle

measure.

AVOID COMMON ERRORSRemind students about the notation used to represent the inverse sine and cosine: si n -1 x and co s -1 x. As with the inverse tangent, the –1 is not an exponent. Rather, it denotes the inverse trigonometric ratio whose value is an angle with a degree measure.

ELABORATE INTEGRATE TECHNOLOGYHave students graph y = sin x and y = cos x on their graphing calculators on the interval [0°, 90°]. Discuss similarities and differences.



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16. How are the inverse sine and cosine ratios for an acute angle of a right triangle defined?

17. Essential Question Check-In How do you find an unknown angle measure in a right triangle?

Evaluate: Homework and Practice

• Online Homework• Hints and Help• Extra Practice

Write each trigonometric expression. Round trigonometric ratios to the nearest thousandth.

1. Given that sin 60° ≈ 0.866, write the cosine of a complementary angle.

2. Given that cos 26° ≈ 0.899, write the sine of a complementary angle.

Write each trigonometric ratio as a fraction and as a decimal, rounded (if necessary) to the nearest thousandth.

3. sin A

4. cos A

5. cos B

6. sin D

7. cos F

8. sin F

A

13

5

12

C

B

D

257

24E

F

Because the sine ratio for a given acute angle is always the same, the measure of that angle

can be defined as an inverse sine ratio:

sin A = BC _ AB

→ m∠A = si n -1 ( BC _ AB

) or sin A = x → m∠A = sin -1 x

Similarly,

cos A = AC _ AB

→ m∠A = c os -1 ( AC _ AB

) or cos A = y → m∠A = cos -1 y

First, use two known side lengths to form a ratio. Then use the appropriate inverse

trigonometric ratio to find the angle measure.

sin 60° = cos (90° - 60°)

0.866 ≈ cos 30°cos 26° = sin (90° - 26°)

0.899 ≈ sin 64°

sin A = opposite

__ hypotenuse

= BC _ AB

= 12 _ 13

≈ 0.923

cos A = adjacent

__ hypotenuse

= AC _ AB

= 5 _ 13

≈ 0.385

cos B = adjacent

__ hypotenuse

= BC __ AB

= 12 _ 13

≈ 0.923

sin D = opposite

__ hypotenuse

= EF _ DF

= 7 _ 25

= 0.28

cos F = adjacent

__ hypotenuse

= EF _ DF

= 7 _ 25

= 0.28

sin F = opposite

__ hypotenuse

= 24 _ 25

= 0.96




QUESTIONING STRATEGIESHow are cosine and the inverse cosine related? The cosine of an acute angle of a

right triangle is the ratio of the lengths of the

adjacent side to the hypotenuse of the right

triangle. The inverse cosine is the angle with that

cosine ratio.

SUMMARIZE THE LESSONHow do you decide which trigonometric ratio to use to find a missing side length or angle

measure in a right triangle? Use sine or inverse sine

for opposite side and hypotenuse relationships and

cosine or inverse cosine for adjacent side and

hypotenuse relationships.

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Find the unknown length x in each right triangle, to the nearest tenth.

9. 10.

11. 12.

Find each acute angle measure, to the nearest degree.

13. m∠P 14. m∠Q

15. m∠U 16. m∠W

C

15

37°

ABx

E

D

27

53°x

F

75°14

K

xL

J

24°

19

R

QxP

8 18

QR

P 11

15

W

U

V

sin C = AB _ AC

sin 37° = x _ 15

⇒ 9.0 ≈ x

cos E = EF _ DE

cos 53° = x _ 27

⇒ 16.2 ≈ x

cos L = JL _ KL

cos 75° = 14 _ x ⇒ ≈ 54.1 sin P = PR _ PQ

sin 24° = 19 _ x ⇒ ≈ 46.7

cos P = PR _ PQ

= 8 _ 18

m∠P = cos -1 ( 8 _ 18

) ≈ 64°

sin U = VW _ UW

= 11 _ 15

m∠U = sin -1 ( 11 _ 15

) ≈ 47°

m∠Q = 90° - m∠P ≈ 90° - 64° = 26°

m∠W = 90° - m∠U ≈ 90° - 47° = 43°


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IN2_MNLESE389847_U7M18L2 948 18/04/14 11:32 PMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1–16 1 Recall of Information MP.4 Modeling

17–24 2 Skills/Concepts MP.4 Modeling

25 3 Strategic Thinking MP.3 Logic

26 3 Strategic Thinking MP.2 Reasoning

27 3 Strategic Thinking MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreInvestigating Ratios in a Right Triangle

Exercise 17

Example 1Finding the Sine and Cosine of an Angle

Exercises 3–8

Example 2Using Complementary Angles

Exercises 1–2

Example 3Finding Side Lengths using Sine and Cosine

Exercises 9–12

Example 4Finding Angle Measures using Sine and Cosine

Exercises 13–16

AVOID COMMON ERRORSRemind students that their calculators must be set on degrees, not radians, to get the correct values for the trigonometric ratios.

Communicating MathDiscuss the importance of being able to draw and interpret right triangle diagrams to use trigonometric ratios to solve real-world problems.



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17. Use the property that corresponding sides of similar triangles are proportional to explain why the trigonometric ratio sin A is the same when calculated in △ADE as in △ABC.

18. Technology The specifications for a laptop computer describe its screen as measuring 15.6 in. However, this is actually the length of a diagonal of the rectangular screen, as represented in the figure. How wide is the screen horizontally, to the nearest tenth of an inch?

19. Building Sharla’s bedroom is directly under the roof of her house. Given the dimensions shown, how high is the ceiling at its highest point, to the nearest tenth of a foot?

20. Zoology You can sometimes see an eagle gliding with its wings flexed in a characteristic double-vee shape. Each wing can be modeled as two right triangles as shown in the figure. Find the measure of the angle in the middle of the wing, ∠DHG to the nearest degree.

DB

E

C

A

R

QP

15.6 in

29°

E

DC

B

A

10.6 ft

5.5 ft

roof

22°

E

G

D

F

H

38 cm

49 cm

43 cm

32 cm

∠A ≅ ∠A and ∠ABC ≅ ∠ADE, since both are right angles.

By AA ~, △ABC~△ADE, so corresponding sides are proportional:

AC _ AE

= BC _ DE

⇒ AC = (BC) (AE)

_ DE

⇒ AC _ BC

= AE _ DE

These ratios determine sin A, and since they are equal, sin A is the same

when calculated in either right triangle.

cos P = PQ

_ PR

⇒ cos29° = PQ

_ 15.6

⇒ 13.6 in. = PQ

sin (m∠AEB) = AB _ AE

⇒ sin 22° = AB _ 10.6

⇒ AB ≈ 4.0

Maximum height: AC = AB + BC ≈ 4.0 + 5.5 = 9.5 ft

cos (m∠DHE) = EH _ DH

= 32 _ 49

m∠DHE = cos -1 ( 32 _ 49

) ≈ 49.2°

sin (m∠FHG) = FG _ GH

= 38 _ 43

m∠FHG = sin -1 ( 38 _ 43

) ≈ 62.1°

m∠DHG = m∠DHE + m∠FHG ≈

49.2 + 62.1 = 111.3°



IN2_MNLESE389847_U7M18L2 949 18/04/14 11:32 PM

AVOID COMMON ERRORSStudents may have difficulty solving equations such as sin x = a __ c when the variable is in the denominator. Review with students how to use inverse operations to write an equivalent equation with the variable in the numerator, such as c = a _____ sin x.

INTEGRATE TECHNOLOGYIn addition to calculators, students can use geometry software and spreadsheets to evaluate trigonometric ratios.

949 Lesson 18 . 2


21. Algebra Find a pair of acute angles that satisfy the equation sin (3x + 9) = cos (x + 5) . Check that your answers make sense.

22. Multi-Step Reginald is planning to fence his back yard. Every side of the yard except for the side along the house is to be fenced, and fencing costs $3.50/yd. How much will the fencing cost?

23. Architecture The sides of One World Trade Center in New York City form eight isosceles triangles, four of which are 200 ft long at their base BC. The length AC of each sloping side is approximately 1185 ft.

Find the measure of the apex angle BAC of each isosceles triangle, to the nearest tenth of a degree. (Hint: Use the midpoint D of

_ BC to create two

right triangles.)

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13 yd32°

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B

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200 ft

1185 ft

C

The expressions must be the measures of two complementary angles:

(3x + 9) + (x + 5) = 90 ⇒ x = 19

(3x + 9) ° = (3 (19) + 9) ° = 66° and (x + 5) ° = ( (19) + 5) ° = 24°

Check: 66° + 24° = 90°

sin (m∠LKM) = LM _ LK

⇒ sin 32° = (23 - 13)

_ LK

⇒ LK ≈ 18.9

distance to fence: JK + KL + LN ≈ 13 + 18.9 + 23 = 54.9 yd

cost of fencing: about 54.9 (3.50) = $192.15

In △ABD, AB = 1185 ft and BD = 1 _ 2

(200 ft) = 100 ft. Therefore,

sin (m∠BAD) = BD _ AB

= 100 _ 1185

⇒ m∠BAD = sin -1 ( 100 _ 1185

)

So m∠BAC = 2m∠BAD = 2 sin -1 ( 100 _ 1185

) ≈ 9.7°.





COGNITIVE STRATEGIESDiscuss how students can use the facts that sin 0° = 0 and the sine increases as the angle increases (between 0° and 90°), and cos 0° = 1 and the cosine decreases as the angle increases (between 0° and 90°), as a quick check when they evaluate sine and cosine trigonometric ratios.



H.O.T. Focus on Higher Order Thinking

24. Explain the Error Melissa has calculated the length of _ XZ in △XYZ. Explain why

Melissa’s answer must be incorrect, and identify and correct her error.

Melissa’s solution:

cos X = XZ _ XY

XZ = XY _ cos X

XZ = 27 cos 42° ≈ 20.1

25. Communicate Mathematical Ideas Explain why the sine and cosine of an acute angle are always between 0 and 1.

26. Look for a Pattern In △ABC, the hypotenuse _ AB has a length of 1.

Use the Pythagorean Theorem to explore the relationship between the squares of the sine and cosine of ∠A, written s in 2 A and c os 2 A. Could you derive this relationship using a right triangle without any lengths specified? Explain.

27. Justify Reasoning Use the Triangle Proportionality Theorem to explain why the trigonometric ratio cos A is the same when calculated in △ADE as in △ABC.

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Com

pan

y

Z

YX27

42°

A

B

C

A

B

1

C

DB

E

C

A

20.1 < 27, but the hypotenuse should

be the longest side. Her definition of

cosine was inverted.

cos X = XY _ XZ

⇒ XZ = 27 _ cos 42°

≈ 36.3

0 < BC < AB ⇒ 0 _ AB

< BC _ AB

< AB _ AB

⇒ 0 < sinA < 1

The same argument shows that 0 < cos A < 1.

sin A = BC ___

AB = BC

___ 1 = BC and cos A = AC ___

AB = AC __ 1 = AC

By the Pythagorean Theorem, A C 2 + BC 2 = AB 2

(sin A) 2 + (cos A) 2 = 1 2

sin 2 A + cos 2 A = 1

Yes: sin 2 A + cos 2 A = ( BC _ AB

) 2

+ ( AC _ AB

) 2

= BC 2 + AC 2

_ AB 2

= AB 2 _ AB 2

= 1

Two segments ⊥ to the same line are ∥ to each other,

so ̄ BC ∥ ̄ DE . By the Triangle Proportionality Theorem, ̄ BC

divides sides ̄ AD and ̄ AE of △ADE proportionally:

BD __ AB = CE __ AC ⇒ 1 + BD __ AB = 1 + CE __ AC ⇒ AB + BD ______ AB = AC + CE

______ AC ⇒ AD __ AB = AE __ AC ⇒ AD __ AE = AB __ AC

These ratios determine cos A, and since they are equal, cos A is the same when

calculated in either triangle.

In △ABD, AB = 1185 ft and BD = 1 _ 2

(200 ft) = 100 ft. Therefore,

sin (m∠BAD) = BD _ AB

= 100 _ 1185

⇒ m∠BAD = sin -1 ( 100 _ 1185

)

So m∠BAC = 2m∠BAD = 2 sin -1 ( 100 _ 1185

) ≈ 9.7°.




PEER-TO-PEER DISCUSSIONHave students investigate the following identities by evaluating them for different angles: si n -1 (sin x) = x and co s -1 (cos x) = x.

JOURNALHave students describe how they remember the difference between the sine and cosine ratios, and the relationship between the ratios. Remind them to include at least one figure and at least one example in their descriptions.

951 Lesson 18 . 2


Lesson Performance TaskAs light passes from a vacuum into another medium, it is refracted—that is, its direction changes. The ratio of the sine of the angle of the incoming incident ray, I, to the sine of the angle of the outgoing refracted ray, r, is called the index of refraction:

n = sin I ___ sin r . where n is the index of refraction.

This relationship is important in many fields, including gemology, the study of precious stones. A gemologist can place an unidentified gem into an instrument called a refractometer, direct an incident ray of light at a particular angle into the stone, measure the angle of the refracted ray, and calculate the index of refraction. Because the indices of refraction of thousands of gems are known, the gemologist can then identify the gem.

1. Identify the gem, given these angles obtained from a refractometer:

a. I = 71°, r = 29°

b. I = 51°, r = 34°

c. I = 45°, r = 17°

2. A thin slice of sapphire is placed in a refractometer. The angle of the incident ray is 56°. Find the angle of the refracted ray to the nearest degree.

3. An incident ray of light struck a slice of serpentine. The resulting angle of refraction measured 21°. Find the angle of incidence to the nearest degree.

4. Describe the error(s) in a student’s solution and explain why they were error(s):

n = sin I _ sin r

= sin 51° _ sin 34°

= 51° _ 34°

= 1.5 → coral

Gem Index of Refraction

Hematite 2.94

Diamond 2.42

Zircon 1.95

Azurite 1.85

Sapphire 1.77

Tourmaline 1.62

Serpentine 1.56

Coral 1.49

Opal 1.39©

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gem

refracted ray

incident ray

4. Possible answer: The student divided out the word “sin” from the

numerator and the denominator in Step 2. This was incorrect because

“sin 51°” and “sin 34°” are indivisible units, with each representing a

single real number.

1. a. sin (71°) ______

sin (29°) = 1.95 zircon

b. sin (51°) ______

sin (34°) = 1.39 opal

c. sin (45°) ______

sin (17°) = 2.42 diamond

2. 1.77 = sin (56°) ______

sin (x°)

sin (x°) = sin (56°) ______ 1.77

x = si n -1 ( sin (56°) ______ 1.77 ) = 28°

3. 1.56 = sin (x°) ______

sin (21°)

x = si n -1 (1.56 ⋅ sin (21°) ) =34°




EXTENSION ACTIVITY

Physicists define the index of refraction of a material in terms of the speed of light. Research the formula used by physicists that relates the index of refraction of a substance, n, the speed of light in a vacuum, c, and the speed of light through the substance, v. n = c __ v

The speed of light in azurite is approximately 100,693 miles per second. Use that fact and the table in the Lesson Performance Task to find the speed of light in a vacuum. Express your answer in miles per second and miles per hour. 186,282 mi/sec; 670,615,200 mi/hr

The Sun is about 93 million miles from Earth. How long does it take light from the Sun to reach Earth? about 8.3 min

AVOID COMMON ERRORSQuestion 4 in the Lesson Performance Task identifies an error commonly made by students in working with the trigonometric functions. Here is a similar error:sin 50° ______ sin 10° = sin 50° ____

10° = sin 5° ≈ 0.0872

Explain to students that the function and the angle cannot be separated. The correct solution is:

sin 50° ____ sin 10° ≈ 0. 7660 _______ 0.1736 ≈ 4.41

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Using a refractometer, a gemologist found that the index of refraction of a substance was 2.0. Using the same angle of incidence, the gemologist found that the angle of the refracted ray for a second substance was greater than the angle of the refracted ray for the first substance. Was the index of refraction of the second substance greater or less than 2.0? Explain. Sample answer: Let i = angle of incidence,

r 1 = refraction angle 1, and r 2 = refraction angle 2.

Then, 2 = sin i _____ si n r 1

. Since the sine function increases as

an angle increases from 0° to 90°, sin r 2 > sin r 1

and sin i _____ sin r 2

< sin i _____ si n r 1

= 2. So, sin i _____ si n r 2

, the index of

refraction of the second substance, was less than 2.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



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