CHAPTER Vocabulary 8 · through the chapter, ... and a clarifying example. angle of depression...

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Vocabulary8

CHAPTER

The table contains important vocabulary terms from Chapter 8. As you workthrough the chapter, fill in the page number, definition, and a clarifying example.

angle of depression

angle of elevation

componentform

cosine

equal vectors

geometric mean

Term Page Definition Clarifying Example


Vocabulary8

CHAPTER

The table contains important vocabulary terms from Chapter 8. As you workthrough the chapter, fill in the page number, definition, and a clarifying example.

angle of depression

angle of elevation

componentform

cosine

equal vectors

geometric mean

544

544

559

525

561

519

The angle formed by ahorizontal line and aline of sight to a pointbelow.

The angle formed by ahorizontal line and aline of sight to a pointabove.

The form of a vectorthat lists the verticaland horizontal changefrom the initial point tothe terminal point.

In a right triangle, thecosine of angle A is theratio of the length ofthe side adjacent toangle A to the length ofthe hypotenuse. It isthe reciprocal of thesecant function.

Two vectors that havethe same magnitudeand the same direction.

For positive numbers aand b, the positivenumber x such that �ax� � �b

x�. In a geometric

sequence, a term thatcomes between twogiven nonconsecutiveterms of the sequence.

The component form ofCD�� is �2.3�.


Angle ofdepression

Angle ofelevation

adjacentleg

hypotenuse

A

C 2

3

D

Given: 4 and 9

Let x be the geometricmean.x2 � (4)(9) � 36x � 6


CHAPTER 8 VOCABULARY CONTINUED

magnitude

parallel vectors

resultant vector

sine

tangent of anangle

trigonometricratio

vector



CHAPTER 8 VOCABULARY CONTINUED

magnitude

parallel vectors

resultant vector

sine

tangent of anangle

trigonometricratio

vector

560

561

561

525

525

525

559

The length of a vector,written �AB�� or �v�� .

Vectors with the sameor opposite direction.

The vector thatrepresents the sum oftwo given vectors.

In a right triangle, theratio of the length ofthe side opposite �A tothe length of thehypotenuse.

In a right triangle, theratio of the length ofthe side opposite �A tothe length of the legadjacent to �A.

A ratio of two sides of aright triangle.

A quantity that has bothmagnitude anddirection.

The sine of an angle isthe ratio of the lengthof the leg opposite theangle to the length ofthe hypotenuse.


� �4, �2�� (4 � 0�)2 � (��2 � 0�)2�� 20� 4.5

parallel vectors

u

resultantvector

v

u + v

�

� �

�

oppositeleg

hypotenuse

A

oppositeside

Aadjacent leg

4

–2

2y

x0

(4, –2)

C 2

3

D

Chapter Review8

CHAPTER


8-1 Similarity in Right Triangles

Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form.

1. 10 and 12 2. 4.5 and 32 3. �54� and �

190�

Find x, y, and z.

4. 5. 6.

7. A surveyor needs to find the distance across a pond.What is AD to the nearest tenth of a meter?

8-2 Trigonometric Ratios

Use a special right triangle to write each trigonometric ratio as a fraction.

8. tan 60° 9. cos 60° 10. sin 60°

Use your calculator to find each trigonometric ratio. Round to the nearesthundredth.

11. sin 12° 12. cos 59° 13. tan 17°

B

AD

C

42 m

15 m

y z12

24 x

z

x

y36y

2 8

x z

18�5�

Chapter Review8

CHAPTER


8-1 Similarity in Right Triangles

Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form.

1. 10 and 12 2. 4.5 and 32 3. �54� and �

190�

Find x, y, and z.

4. 5. 6.

7. A surveyor needs to find the distance across a pond.What is AD to the nearest tenth of a meter?

8-2 Trigonometric Ratios

Use a special right triangle to write each trigonometric ratio as a fraction.

8. tan 60° 9. cos 60° 10. sin 60°

Use your calculator to find each trigonometric ratio. Round to the nearesthundredth.

11. sin 12° 12. cos 59° 13. tan 17°

0.310.520.21

�12��3�

48.9 m

B

AD

C

42 m

15 m

x � 6, y � 12�5�, z � 6�5�x � 9, y � 18, z � 9�5�x � 4; y � 2�5�; z � 4�5�

y z12

24 x

z

x

y36y

2 8

x z

�5�

62�

�122�30�

�3��

2

18�5�


Find each length. Round to the nearest hundredth.

14. LK 15. AB 16. WX

8-3 Solving Right Triangles

Find the unknown measures. Round lengths to the nearest hundredthand angle measures to the nearest degree.

17. 18. 19.

20. A bike trail has a sign warning bikers of a 7° inclinefor the next 12 miles. To the nearest tenth of a mile, find the vertical distance of the incline, h.

8-4 Angles of Elevation and Depression

21. From the top of a lighthouse to a ship the angle of depression is 28°. The top of the lighthouse is located 175 feet above the water. What is the horizontal distance from the lighthouse to the ship? Round to the nearest foot.

28

175 ft

7°

12 mih

32°

9.5

D E

F

3.6

5.5 ML

N

25° BC

A

14

51°xw

y

9.6

40°C

A

B

10

35°

J

L K

20

CHAPTER 8 REVIEW CONTINUED


Find each length. Round to the nearest hundredth.

14. LK 15. AB 16. WX

8-3 Solving Right Triangles

Find the unknown measures. Round lengths to the nearest hundredthand angle measures to the nearest degree.

17. 18. 19.

20. A bike trail has a sign warning bikers of a 7° inclinefor the next 12 miles. To the nearest tenth of a mile, find the vertical distance of the incline, h.

8-4 Angles of Elevation and Depression

21. From the top of a lighthouse to a ship the angle of depression is 28°. The top of the lighthouse is located 175 feet above the water. What is the horizontal distance from the lighthouse to the ship? Round to the nearest foot.

329 feet

28

175 ft

7°

12 mih

1.5 mi

�F � 58˚; DE � 8.06;DF � 5.03

�L � 33˚; �N � 57;LN � 6.57

�A � 65˚; AB � 33.13;CB � 30.02

32°

9.5

D E

F

3.6

5.5 ML

N

25° BC

A

14

6.046.4328.56

51°xw

y

9.6

40°C

A

B

10

35°

J

L K

20



22. The angle of elevation of the sun is 82°, a water tower casts a shadow that is 22 m long. What is the height of the water tower to the nearest tenth of a meter?

8-5 Law of Sines and Law of Cosines

Find each measure. Round lengths to the nearest tenth and anglemeasures to the nearest degree.

23. m�P 24. AB 25. AC

26. JL 27. m�A 28. RP

29°

12

15

P

R Q16

19 12

A C

B

59°

13

9

JK

L

34°71°

115

C

AB

103°24°

73C

A

B42°

43

27P

Q

R

82



22. The angle of elevation of the sun is 82°, a water tower casts a shadow that is 22 m long. What is the height of the water tower to the nearest tenth of a meter?

8-5 Law of Sines and Law of Cosines

Find each measure. Round lengths to the nearest tenth and anglemeasures to the nearest degree.

23. m�P 24. AB 25. AC

26. JL 27. m�A 28. RP

7.439˚11.4

29°

12

15

P

R Q16

19 12

A C

B

59°

13

9

JK

L

66.659.825˚

34°71°

115

C

AB

103°24°

73C

A

B42°

43

27P

Q

R

156.5 m

82



8-6 Vectors

Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.

29. �4, 1� 30. ��3, �5� 31. �0, 6�

Draw each vector on a coordinate plane. Find the direction of the vector to thenearest degree.

32. A wind velocity is given by 33. The path of a hiker is given by the vector �2, 4�. the vector �7, 4�.

34. To reach an island, a ship leaves port and sails for 10 km at a bearing of N 48° E. It then sails due east for 14 km.What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree.

48

14 km Island10 km

W E E E

S

N

2

–2

4

6

y

62 4x

y

x2 4–2

2

–2

4

6

–4

y

x2 4–2

2

–2

4

6

–4

2 4–2

2

–2

4

y

x

–4

–42 4–2

2

–2

4

y

x

–4

–4



8-6 Vectors

Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.

29. �4, 1� 30. ��3, �5� 31. �0, 6�

Draw each vector on a coordinate plane. Find the direction of the vector to thenearest degree.

32. A wind velocity is given by 33. The path of a hiker is given by the vector �2, 4�. the vector �7, 4�.

34. To reach an island, a ship leaves port and sails for 10 km at a bearing of N 48° E. It then sails due east for 14 km.What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree.

22.5 km; N73 E

48

14 km Island10 km

W E E E

S

N

30°

2

–2

4

6

y

62 4x

63°y

x2 4–2

2

–2

4

6

–4

6�34� 5.8�17� 4.1

y

x2 4–2

2

–2

4

6

–4

2 4–2

2

–2

4

y

x

–4

–42 4–2

2

–2

4

y

x

–4

–4



Theorem 8-1-1 The altitude to the hypotenuse of a right triangle forms twotriangles that are similar to each other and to the originaltriangle.

Corollary 8-1-2 The length of the altitude to the hypotenuse of a righttriangle is the geometric mean of the lengths of the twosegments of the hypotenuse.

Corollary 8-1-3 The length of a leg of a right triangle is the geometric meanof the lengths of the hypotenuse and the segments of thehypotenuse adjacent to that leg.

Theorem 8-5-1 (The Law of Sines) For any �ABC with side lengths a, b,and c,

�sin

aA

� � �sin

bB

� � �sin

cC

�.

Theorem 8-5-2 (The Law of Cosines) For any �ABC with side lengths a, b,and c,

a2 � b2 � c2 � 2bc cos A,

b2 � a2 � c2 � 2ac cos B, and

c2 � a2 � b2 � 2ab cos C.

Postulates and Theorems8

CHAPTER

Big Ideas8

CHAPTER


Answer these questions to summarize the important concepts fromChapter 8 in your own words.

1. Explain the trigonometric ratios.

2. Explain what vectors are and how they are used.

3. Explain what solving a right triangle means.

4. Explain why and how the Law of Sines and Law of Cosines are used.

For more review of Chapter 8:

• Complete the Chapter 8 Study Guide and Review on pages 572–575 ofyour textbook.

• Complete the Ready to Go On quizzes on pages 543 and 569 of yourtextbook.

Big Ideas8

CHAPTER


Answer these questions to summarize the important concepts fromChapter 8 in your own words.

1. Explain the trigonometric ratios.

2. Explain what vectors are and how they are used.

3. Explain what solving a right triangle means.

4. Explain why and how the Law of Sines and Law of Cosines are used.

For more review of Chapter 8:

• Complete the Chapter 8 Study Guide and Review on pages 572–575 ofyour textbook.

• Complete the Ready to Go On quizzes on pages 543 and 569 of yourtextbook.

Answers will vary. Possible answer: The Law of Sines and Cosinesare used to solve oblique triangles, triangles that have no rightangles. The Law of Sines is best used when given one side and twoangles. The Law of Cosines is best used when given three sides ortwo sides and an included angle.

Answers will vary. Possible answer: Solving a right triangle meansto find the measures of all angles and all sides.

Answers will vary. Possible answer: A vector can represent speedand direction. It has both length and direction. It has a horizontaland vertical change from its initial point to its terminal point.

Answers will vary. Possible answer: The trigonometric ratios aresine, cosine and tangent. They are used to solve right triangles.Sine is the opposite side over the hypotenuse. Cosine is theadjacent side over the hypotenuse. Tangent is the opposite sideover the adjacent side.

CHAPTER Vocabulary 8 · through the chapter, ... and a clarifying example. angle of depression...

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Transcript of CHAPTER Vocabulary 8 · through the chapter, ... and a clarifying example. angle of depression...