special right triangles - administration · 2011-03-19 · special right triangles. georgia...

Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 163

Special Right Triangles5.1Goal p Use the relationships among the sides in

special right triangles.GeorgiaPerformanceStandard(s)

MM2G1a, MM2G1b

Your Notes

THEOREM 5.1: 458-458-908 TRIANGLE THEOREM

In a 458-458-908 triangle, the hypotenuse

2x

x

x458

458

is times as long as each leg.

hypotenuse 5 leg p

Find the value of x.

a.6

458

x

b. 29

x x

Solutiona. By the Triangle Sum Theorem, the measure of the

third angle must be . Then the triangle is a - -908 triangle, so by Theorem 5.1, the

hypotenuse is times as long as each leg.

hypotenuse 5 leg p - -908Triangle Theorem

x 5 Substitute.

b. You know that each of the two congruent angles in the triangle has a measure of because the sum of the angle measures in a triangle is 1808.

hypotenuse 5 leg p - -908Triangle Theorem

5 x p Substitute.

5 x Divide each side by .

5 x Simplify.

Example 1 Find lengths in a 458-458-908 triangle

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164 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.

Your Notes

1.

22 22

x458

2.

212

xx

Checkpoint Find the value of x.

THEOREM 5.2: 308-608-908 TRIANGLE THEOREM

In a 308-608-908 triangle, the hypotenuse is as long as the shorter leg, and the longer leg is times as long as the shorter leg.

hypotenuse 5 p shorter leg x 2x

308

608

3xlonger leg 5 shorter leg p

Music You make a guitar pick that resembles an equilateral triangle with side lengths of 32 millimeters. What is the approximate height of the pick?

SolutionDraw the equilateral triangle described.

608

32 mmh32 mm

16 mm 16 mmDA C

B

Its altitude forms the longer leg of two - -908 triangles. The length

h of the altitude is approximately the height of the pick.

longer leg 5 shorter leg p

h 5 p < mm

Example 2 Find the height of an equilateral triangle

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Your Notes

Find the values of x and y. Write your answer in simplest radical form.

xy

308

608

8

SolutionStep 1 Find the value of x.

longer leg 5 shorter leg p

5 x Substitute.

5 x Divide each side by .

p 5 x Multiply numerator and denominator by .

5 x Multiply fractions.

Step 2 Find the value of y.

hypotenuse 5 p shorter leg

y 5 p 5

Example 3 Find lengths in a 308-608-908 triangle

Checkpoint Find the value of the variable.

3.

308

608

x

32

4.

h12 12

66

Homework

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Find the value of x. Write your answer in simplest form.

1.

24x

458

458

2.

x

3458

458

3. 28

xx

458 458

4.

27

x

x

5. 5

5x

6.

4

x

Complete the table.

7.

y

x

x

458

x 3 7 2

y 6 Ï}

2

LESSON

5.1 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON

5.1 Practice continued

8.

ac

b

60º

30º

a 4 6

b 5 Ï}

3

c 8 Ï}

3

Find the value of each variable. Write your answers in simplest form.

9.

x

y

60º

30º

6

10.

60º

30ºx

y 32

11. 60º

30ºx

y

9

12. 7

y 8 y 8

x

13.

xº

30º

y10

14. 60º

y

2x

36



Find the value of each variable. Write your answers in simplest form.

15.

y

60º

xº

10

16.

2 y

y5x°

5x°

17.

2xº

xº

32

y

The side lengths of a triangle are given. Determine whether it is a 458-458-908 triangle, a 308-608-908 triangle, or neither.

18. 2, 4, 2 Ï}

3 19. 5, 5, 5 Ï}

2 20. 6, 12, 8

21. 11, 11, 11 Ï}

2 22. 10, 20, 10 Ï}

3 23. 3, 4, 5

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


24. Construction A construction worker is building a ramp to

30°

8 ft

make transportation of materials easier between an upper and lower platform. The upper platform is 8 feet off the ground, and the angle of elevation is 308. How long is the ramp?

25. Art Gallery A designer is creating a new and unique

20 ft308

20 ft308

entryway for an art gallery. The designer wants the entryway to slant inward at a 308 angle on either side of the wall as shown.

a. How tall should the entryway be if the inner length of the entryway is 20 feet? Round your answer to the nearest foot.

b. How wide is the base of the entryway, from end to end?



5.2 Apply the Tangent RatioGoal p Use the tangent ratio for indirect measurement.Georgia

PerformanceStandard(s)

MM2G2a, MM2G2b, MM2G2c

Your Notes

VOCABULARY

Trigonometry

Trigonometric ratio

Tangent

Complementary angles

TANGENT RATIO

Let nABC be a right triangle B

C A

with acute ∠A. The tangent of ∠A (written as tan A) is defined as follows:

tan A 5length of leg opposite ∠A}}}length of leg adjacent to ∠A 5

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Your Notes

Find the value of x.

25°

x

12

Use the tangent of an acute angle to find a leg length.

tan 258 5 Write ratio for tangent of 258.

tan 258 5 Substitute.

p tan 258 5 x Multiply each side by .

( ) ø x Use a calculator to find tan 258.

ø x Simplify.

Example 2 Find a leg length

1. Find tan A and tan B. Round

30

24 18

A B

C

to four decimal places.

2. Find the value of x. Round to

38°

x

20 AC

B

the nearest tenth.

Checkpoint Complete the following exercises.

Find tan X and tan Y. Write each Y

Z X

8

15

17answer as a fraction and as a decimal rounded to four places.

tan X 5opp. ∠X}adj. to ∠X 5 5 5

tan Y 5opp. ∠Y}adj. to ∠Y 5 5 5

In the right triangle, nXYZ, ∠X and ∠Y are angles. You can see that the

tangent ratios of the angles are .

Example 1 Find tangent ratios

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Your Notes

Find the height h of the flagpole

70°7 ft

h ft

to the nearest foot.

tan 708 5 Write ratio for tangent of 708.

tan 708 5 Substitute.

p tan 708 5 Multiply each side by .

ø h Use a calculator to simplify.

The flagpole is about feet tall.

Example 4 Estimate height using tangent

3. Find the height h of the flagpole in Example 4 to the nearest foot if the angle is 758.

Checkpoint Complete the following exercise.

Homework

Find tan X and tan Y for similar triangles. Then compare the tangent ratios.

c

a

b

Y

Z X

3c

3a

3b

Y

Z X

tan X 5 tan X 5 5

tan Y 5 tan Y 5 5

The values of tan X and tan Y for the similar triangles are .

Example 3 Compare the tangent ratios for similar triangles

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Name ——————————————————————— Date ————————————

LESSON

5.2 PracticeFind tan A and tan B. Write each answer as a decimal rounded to four decimal places.

1. B C

A

1237

35 2. A

CB

29 21

20

3.

AC

B

3

4

5

Find the value of x to the nearest tenth.

4.

x458

15

5.

39°

24

x

6.

32°

9x

7.

34°12

x 8. 67°

18

x

9.

43°24

x



Find tan X and tan Y for the similar triangles. Then compare the ratios.

10. Y

Z X24

2610

Y

Z X12

135

11.

XX

Y

Y

ZZ

33 65

56

13066

112

Find the value of x using the defi nition of tangent. Then fi nd the value of x using the 458-458-908 Triangle Theorem or the 308-608-908 Triangle Theorem. Compare the results.

12.

x

458

7

13. 1030°

x

14. 608

6 3

x

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find the area of the triangle. Round your answer to the nearest tenth.

15.

x

48º17

16.

37°

x

24

17.

63°x

6

18.

53°

x 11

19.

27°

x

28

20.

73° x6

Use the tangent ratio to fi nd the value of x. Round to the nearest tenth.

21.

x

458

16 22.

x

428

35

23. x

57819



24. Billboard You are standing 35 feet from a billboard.

58°35 ft

On

Sale Now!

The angle of elevation from your position to the top of the billboard is 588. How tall is the billboard? Round your answer to the nearest foot.

25. Tree Your friend is standing near a tree that is 18 feet tall.

37°

18 ft

Yourfriend

The angle of depression from the top of the tree to your friend is 378. How far is your friend from the tree? Round your answer to the nearest foot.

LESSON


Name ——————————————————————— Date ————————————



Apply the Sine and Cosine Ratios

5.3Goal p Use the sine and cosine ratios.Georgia


MM2G2a, MM2G2b, MM2G2c

Your Notes

VOCABULARY

Sine

Cosine

SINE AND COSINE RATIOS

Let nABC be a right triangle with B

C A

acute ∠A. The sine of ∠A and cosine of ∠A (written sin A and cos A) are defined as follows:

sin A 5length of leg opposite ∠A}}}

length of hypotenuse 5

cos A 5length of leg adjacent to ∠A}}}

length of hypotenuse 5

Find sin X and sin Y. Write each Y

Z X

725

24

answer as a fraction and as a decimal rounded to four places.

sin X 5opp. ∠X}

hyp. 5 5 5

sin Y 5opp. ∠Y}

hyp. 5 5 5

Example 1 Find sine ratios

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Your Notes

Find cos X and cos Y. Write each Y

Z X

725

24

answer as a fraction and as a decimal rounded to four places.

cos X 5adj. to ∠X}

hyp. 5 5 5

cos Y 5adj. to ∠Y}

hyp. 5 5 5

Example 2 Find cosine ratios

1. Find sin A and sin B. B

C A

2129

20 2. Find cos A and cos B.

Checkpoint Find the indicated measure. Round to 4 decimal places, if necessary.

Use a trigonometric ratio to find the value of x in the diagram. Round to the nearest tenth.

a.x

31°12

b.

x

44°

48

a. cos 318 5 b. sin 448 5

cos 318 5 sin 448 5

x 5 p sin 448 5 x

x ø ( ) ø x

x ø ø x

Example 3 Use trigonometric ratios to find side lengths

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Your Notes

Find the height of the parking ramp shown.

sin 278 5 65 ft

27°x ft

sin 278 5

p sin 278 5 x

( ) ø x

ø x

Example 5 Use trigonometric ratios to find side lengths

3. Find the value of x. Round to

x

y

28

46°

the nearest tenth.

4. Find the value of y. Round to the nearest tenth.


Find the sine and cosine of Y

Z Xab c

M

N L3a

3b3c

∠X, ∠Y, ∠L, and ∠M of the similar triangles. Then compare the ratios.

sin X 5 cos X 5

sin Y 5 cos Y 5

sin L 5 5 cos L 5 5

sin M 5 5 cos M 5 5

In nXYZ, ∠X and ∠Y are angles, so sin X 5 cos and sin Y 5 cos . In nLMN, ∠L and ∠M are angles, so sin L 5 cos and sin M 5 cos . Because nXYZ and nLMN are

triangles, sin X 5 sin , cos X 5 cos ,sin Y 5 sin , and cos Y 5 cos .

Example 4 Sine and cosine ratios for similar triangles

Homework

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Find sin R and sin S. Write each answer as a decimal rounded to four decimal places.

1.

16 20

12

S

T R

2.

32 40

24T

R

S 3. 24

2610

S T

R

Find cos A and cos B. Write each answer as a decimal rounded to four decimal places.

4.

41

9

40

B

A

C

5.

6

B

AC

8 10

6.

39

80C A

B

89

7. Find sin X, sin Y, cos X, and cos Y. Then compare the ratios.

35

1237

XZ

Y

LESSON

5.3 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


8. Find sine and cosine of ∠ X, ∠ Y, ∠ L, and ∠ M. Then compare the ratios.

35

1237

XZ

Y

70

2474

N L

M

Use a sine or cosine ratio to fi nd the value of each variable. Round decimals to the nearest tenth.

9.

ab

24

72° 10. c

d18

63° 11.

r

s 32

52°



Use a sine or cosine ratio to fi nd the value of each variable. Round decimals to the nearest tenth.

12.

b

a

24

42° 13.

c

d50

61° 14.

sr

1733°

Find the unknown side length. Then fi nd sin A and cos A. Write each answer as a decimal rounded to four decimal places.

15.

38

10c

A

BC

16.

9

12 c

A

B

C

17. 45

27a

A

B

C

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


18. Car A car is driving down a hill that is 500 feet long,

32°

d

500 ft

Not drawn to scale

at an angle of elevation of 328. To the nearest foot, what is the vertical distance d covered by the car?

19. Satellite A satellite is launched at an angle of elevation

d

h

72°

Not drawn to scale

of 728 as shown.

a. Suppose the satellite covers a total distance of 8 miles during the launch. What is its vertical height h? Round to the nearest tenth of a mile.

b. The satellite reaches a vertical height of 3 miles. What is the total distance d covered during the launch? Round to the nearest tenth of a mile.



5.4 Solve Right TrianglesGoal p Use inverse tangent, sine, and cosine ratios.Georgia


MM2G2c

Your Notes

VOCABULARY

Solve a right triangle

INVERSE TRIGONOMETRIC RATIOS

Let ∠A be an acute angle.

A

B

CInverse Tangent If tan A 5 x, then tan21 x 5 m∠A. tan21 BC

}AC 5 m∠A

Inverse Sine If sin A 5 y, then sin21 y 5 m∠A. sin21 BC

}AB 5 m∠A

Inverse Cosine If cos A 5 z, then cos21 z 5 m∠A. cos21 AC

}AB 5 m∠A

Use a calculator to approximate the

B

16

20

C

A

measure of ∠A to the nearest tenth of a degree.

Because tan A 5 5 5 ,

tan21 5 m∠A. Using a calculator, tan21 < .

So, the measure of ∠A is approximately .

Example 1 Use an inverse tangent to find an angle measure

1. In Example 1, use a calculator and an inverse tangent to approximate m∠C to the nearest tenth of a degree.

Checkpoint Complete the following exercise.

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Your Notes

Let ∠A and ∠B be acute angles in two right triangles. Use a calculator to approximate the measures of ∠A and ∠B to the nearest tenth of a degree.

a. sin A 5 0.76 b. cos B 5 0.17

Solution

a. m∠A 5 b. m∠B 5

< <

Example 2 Use an inverse sine and an inverse cosine

Solve the right triangle.

238

A C

B

40 ftRound decimal answers to the nearest tenth.

SolutionStep 1 Find m∠B by using the Triangle Sum Theorem.

5 908 1 238 1 m∠B

5 m∠B

Step 2 Approximate BC using a ratio.

5BC}40 Write ratio for .

5 BC Multiply each side by .

< BC Approximate .

< BC Simplify and round answer.

Step 3 Approximate AC using a ratio.

5AC}40 Write ratio for .

5 AC Multiply each side by .

< AC Approximate .

< AC Simplify and round answer.

The angle measures are , , and . The side lengths are feet, about feet, and about

feet.

Example 3 Solve a right triangle

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Your Notes

Homework

2. Find m∠T to the nearest tenth of a degree if cos T 5 0.64.

3. Find m∠D to the nearest tenth of a degree if sin D 5 0.48.

4. Solve a right triangle that has a 508 angle and a 15 inch hypotenuse.


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Name ——————————————————————— Date ————————————

LESSON

5.4 PracticeMatch the trigonometric expression with the correct ratio. Some ratios may be used more than once, and some may not be used at all.

1. sin A 2. cos A 3. tan A

A C

B

8

15

17

4. sin B 5. cos B 6. tan B

A. 8 }

17 B.

15 }

17 C.

17 }

8

D. 17

} 15

E. 8 }

15 F.

15 }

8

Use a calculator to approximate the measure of ∠ A to the nearest tenth of a degree.

7. A

B C30

20

8.

A

B C15

11

9. A B

C

1426

10. C B

A

1016

11. A

B C7

11

12. B C

A

9 22



Solve the right triangle. Round decimal answers to the nearest tenth.

13.

R

P

12

458

14.

P

N

12

17 15. U

S

T

15708

16. V M

D

21

508

17.

R

T

A

16

30

18. EU

M

15

18

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Let ∠ A be an acute angle in a right triangle. Approximate the measure of ∠ A to the nearest tenth of a degree.

19. sin A 5 0.45 20. tan A 5 0.9 21. sin A 5 0.76 22. cos A 5 0.32

23. tan A 5 5.2 24. cos A 5 0.24 25. sin A 5 0.15 26. cos A 5 0.66

27. Multiple Choice Using the diagram to the right, for what

xA C

Bvalue of x does sin A 5 cos A?

A. 30° B. 45°

C. 60° D. none

28. Ladder You lean a 20 foot ladder against a house. The base

20 ft

4 ftu

of the ladder is 4 feet from the wall. What angle u does the ladder make with the ground?



29. Skyscraper You are standing 350 feet away

x 8

350 ft

750 ft

from a skyscraper that is 750 feet tall. What is the angle of elevation from you to the top of the building?

30. Concert You attend a music concert with some

248

stage

45 ftd

friends and sit halfway up the bleachers in the arena. The angle of depression from your horizontal line of sight to the stage is 24°. If your seat is 45 feet above stage level, what is your actual distance d from the stage? Round to the nearest foot.

LESSON


Name ——————————————————————— Date ————————————



Words to ReviewGive an example of the vocabulary word.

Trigonometry

Tangent

Cosine

Solve a right triangle

Inverse sine

Trigonometric ratio

Complementary angles

Sine

Inverse tangent

Inverse cosine

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special right triangles - administration · 2011-03-19 · special right triangles. georgia...

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