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48 (reflection)

49 (dilation) ( ) ( )

. ( )

.

( ) .

.x, x + 17, 25 .7 ) x 25

(

(2-5 ) .

(trignometric functions)

(sine)

(cosine)

(tangent)

(cosecant)

(secant)

(cotangent)

(reciprocal function)

(inverse sine)

(inverse cosine)

(inverse tangent)

(radian)

(coterminal angles)

(reference angle)

(unit circle)

(circular function)

(period)

(sinusoid)

(amplitude)

(frequency)

(phase shift)

(Law of Sines)

(Law of Cosines)

137

UAE_Math_Grade11_ADVM_S_C03_v2_718372.indb 137 م 08:46:40 03/11/2016

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. 1

.

2

.

.

.

.

.

(trigonometric ratios)

trigonometric)(functions

(sine)

(cosine)

(tangent)

(cosecant)

(secant)

(cotangent)

(reciprocal function)

inverse trigonometric)(function

(inverse sine)

(inverse cosine)

(inverse tangent)

(angle of elevation)

(angle of depression)

(solve a right triangle)

1 .

.

. θ

θ

θ opp, adj, hyp

. θ

: θ

cosecant

secant

cotangent

sine

cosine

tangent

cotangent secant cosecant

. . tangent cosine sine

: cotangent tangent cosine sine

83

cot θ = cos θ _ sin θ

tan θ = sin θ _ cos θ

1-3 | 138


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cosine, sine

.tangent

!

2

10 4

cos θ = 2 _

5

.

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139

1

.θ

15 θ 8 θ

.17

sin θ = opp

_ hyp

= 8

_ 17

opp = 8 hyp = 17 csc θ = hyp

_ opp

= 17

_ 8

cos θ = adj

_ hyp

= 15

_ 17

adj = 15 hyp = 17 sec θ = hyp

_ adj

= 17

_ 15

tan θ = opp

_

adj =

8

_ 15

ppo = 8 jda = 5 cot θ = adj

_ opp

= 15

_ 8

1A. 1B.

: sin θ

.θ

.

. . sin θ

.

2

.θ cos

.θ

.5 2 cos θ = adj

_ hyp

= 2

_ 5

. = θ

.θ θ = 1

_ 2 .2

==


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DEGREE

. MODE

.

.60° 45° 30° :

.45°-45°-90° 30°-60°-90°

.57–62

2 .

3

. .x

. cosine

cosine

θ = 42°, adj = x, hyp = 18

.18

.

13.4 x

.cos 42° x = 13.4

cos 42° =

cos 42° = x = 13.4

0.74 = 0.74 ✔ .

3A. 3B.

| 3-1 | 140


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sin -1 x

.sinx

:

.sin -1 x ≠ 1 _

sin x

.f -1( x )

2.4

. 26.2 112

:

. .

. sine

sin θ = opp

_ hyp

sine

sin 63° = 200

_ x θ = 63°, opp = 200, hyp = x

x sin 63° = 200 .x

224.47 .sin 63°

. 224.5

.4 .

.

4

.

sine .θ x sine x sin θ θ

.sin -1 x = θ sin θ = x

cosine .θ x cosine x cos θ θ

.cos -1 x = θ cos θ = x

tangent .θ x tangent x tan θ θ

.tan -1 x = θ tan θ = x

5

. .θ

.tan θ

tan θ = opp

_

adj tan

tan θ = 26

_ 11

opp = 26, adj = 11

67° θ = tan -1 26

_ 11

tan

5A.

5B.

141


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.

. .

.

. 6 32°

. 144 150 - 6 6

.x

tan θ = opp

_ adj

tan

θ = 32°, opp = 144, adj = x

x tan 32° = 144 .x

.tan 32°

x ≈ 230.4 .

. 230.4

. 1500 .6

. 36°

6

/

.

.

| 3-1 | 142


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.

tan(26

tan(26)

tan(26 + 50

.tan(26) + 50

.

7

52° 28°

. 650 .

.

.

.y x

tan

.x

tan θ = opp

_ adj

tan

tan 52° = 650

_ x θ = 52°, opp = 650, adj = x

x tan 52° = 650 .x

x = 650

_ tan 52°

.tan 52°

.y

tan θ = opp

_ adj

tan

tan 28° = 650

_ x + y θ = 28°x + y, opp = 650, adj = x + y

(x + y) tan 28° = 650

x + y = 650

_ tan 28°

.tan 28°

.x = 650

_ tan 52°

. 650

_ tan 52°

y ≈ 714.6 .

. 714.6

.48° .7 64° 22

.

.x + y

143


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.

.

.

8

.

.

a.

. z x

.

.

.

10 tan 35° = x

7.0 ≈ x

.90° X Y

Y = 90° + 35° . Y X

Y = 55° .

.z ≈ 12.2 Y = 55° x ≈ 7.0

b.

.13.6

. J

tan J = 11 _

8 .

. J = tan

J ≈ 53.97° .

.90° J L J

53.97° + L ≈ 90° . L J

L ≈ 36.03° .

.k ≈ 13.6 J ≈ 54° L ≈ 36° :

8A. 8B.

cos 35°= 10 _

z .

z cos 35° = 10 .

.

z ≈ 12.2 .

| 3-1 | 144


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.θ

(1 )

1. 2.

3. 4.

5. 6.

7. 8.

θ

(2 ) .θ

. .x

(3 )

19. 20.

21. 22.

23. 24.

25. 26.

27

25 .

(4 ) 35°

3.5 .28

(4 ) .18°

. .a

. .b

0.8 A .29

(4 ) .32°

. .a

. A .b

.30

1350

.8°

(4 )

(5 ) . .θ

31. 32.

33. 34.

35. 36.

37. 38.

145


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. .39

800 .

.32°

(6 )

.40

. 135

58°

(6 )

.41

55° 375

(6 ) .

. .a

. .b

.42

225

(6 ) .48°

. .a

.b.

. 10 5 .43

29° 10

.43°

(7 )

.44

.1.4°

219

(7 ) .6.8°

. .a

. .1b

. 156 .45

27°

(7 ) .7°

. .a

. .b

.46

. 60

.44.76° 48°

(7 ) .

.

(8 ) .

47. 48.

49. 50.

52.

. 65 .55

. 10

. .a

.b

.56

1.4

. .a

.b

B

51.

53. 54.

| 3-1 | 146


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.

57. sin 60° 58. cot 30° 59. sec 30°

60. cos 45° 61. tan 60° 62. csc 45°

θ

.

63. tan θ = 1 64. cos θ =

65. cot θ = 66. sin θ =

67. csc θ = 2 68. sec θ = 2

.x

69. 70.

20 .71

.70°

45

.57°

.

θ cos θ

.

72. 3-4-5 73. 5-12-13

.74

.

= > <

.

75. sin 45° cot 60° 76. tan 60° cot 30°

77. cos 30° csc 45° 78. cos 30° sin 60°

79. sec 45° csc 60° 80. tan 45° sec 30°

d .81

.3.5°

.82

.

. P(x, y) .a . P

P, (x, 0)

.

r .y x

.x θ

.y x r .b

x, y tan θ sin θ, cos θ .c

.r /

.d(cos θ, sin θ) P

θ .e

.f

.θ a

θ .83

.cot θ = cos θ

_ sin θ

tan θ = sin θ

_ cos θ

sin θ = a .84

.csc θ

. .

.85

.

θ .86

. .

θ .88 .(sin θ) 2 + (cos θ) 2 = 1

m∠A < m∠B B A .

.

88. sin A < sin B

89. cos A < cos B

90. tan A < tan B

.91

.

.

C

147


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. (CPI) .92

.100 1982–1984

(CPI ) .2005 1955 (CPI) .2025 CPI .1955 x = 0 .

. :

93. e 5x = 24 94. 2e x - 7 - 6 = 0

. .

95. f (x) = -3 x - 2 96. f (x) = 2 3x - 4 + 1 97. f (x) = -4 -x + 6

:

98. 99. 100.

. .101

2002 2003 2004 2005 2006 2007 2008

904.3 814.7 773.9 725.5 716.2 699.1 673.0 ( )

. .2001 x .a

. .b

.2015 .c

CPI

195526.8

196531.5

197553.8

1985107.6

1995152.4

2005195.3

:

z SAT/ACT .102

A 15 D

B E

C

.103

3 . 10

F 9.39 ft

G 10.44 ft

H 11.23 ft

J 12.05 ft

.104

.

.

A 7.8 ft

B 10.5 ft

C 12.9 ft

D 14.3 ft

.45° .105

. 120

F 60 ft

G

H

J 120 ft

| 3-1 | 148


.

1

.

2

.

3-1

.

540

540°

.

1 . .

x .

.

.

.

(°)

( )

.

90° 180° 1 360°

.

(vertex)

(initial side)

(terminal side)

(standard position)

(radian)

(coterminal angles)

(linear speed)

(angular speed)

(sector)

McGra

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149


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DMS

DMS

.Angle

60

(60)

(10)

.

(DMS)

.( ") 60 ( ') 60

DMS 1

DMS ( ) DMS

.

. 0.735°

0.1'

.56° 44' 6'' 56.735°

. .

. 32.091° 32° 5' 28''

.

. .

.

θ

. r s

.s = r θ = 1

θ

1

.

r s

. rad .

| 3-2 | 150


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: .2πr

. π = 180° 2π = 360° 360° 2π

:

:

. .1

. .2

2

. π

1° ≈ 0.017 rad

.1 rad ≈ 57.296°

.

.(°)

151


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( ) α

.( ) θ ( ) β

β α . .

.

2π 360°

.

α

n α + 360n °

.α

α

n α + 2n π

.α

3

.

.

45° + 360n°

.45°

.n = 1 -1

.

.n = 1 -1

| 3-2 | 152


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4a

r = 5

-

.

.

2 s .

r θ

: s

s = rθ

. θ

. - θ

4

. .

. 3.9

. s = rθ 60° 1

. r = 2

. 2

. 2.1

153


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ω

.

.

. .

.

.r

t s

. v

t ( ) θ

. ω

.

140 .a

.

.

140 2π

. 280π 140 × 2π θ

. 879.6 280π

. 2.5 .b .

.5π 2.5 × 2π θ 2.5

.

. 13.4

. DVD

DVD .5A

. 3.5

DVD .5B

. 3

5

35 30

. 45 30

:

75π

1

60

1

60

1

1

12

1

5280

13.4

| 3-2 | 154


.

.P

. A .

.r θ

θ r A

. θ

6

. .a

. 3 θ

. 12.4

.b

. 26

. 10 26 - 16 26

.

= A 1 . . 10 = A 2 26

.

.

. 653.5

.r θ

.67°

12

. 30

Car and Driver :

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155


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) DMS

DMS (

(1 ) .

.9

10

.

(1 ) .

π

(2 ) .

.

(3 ) .

20 . .26

.

. .

(3 )

. .

(4 )

3024° .33

(4 ) .

13 .a

18 .b

a

(5 ) .

.40

(5 ) .

(in.)

(rps)

32800

55500

4500

5500

5000

. .a

.

6 .b

3

.41

. 26 . 840 646

(5 )

. .a

. .b

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. 8 .42

2.4

(5 ) . 3.4 3.2

.a

.

20 .b

(6 ) .

. .49

18

(6 )

. .50

6

(6 )

.

.

θ 2π 0 .55

:

III .c I .a

IV .d II .b

.56

.

.

.57

.

.41° 12' N 33° 26' N

3963

. θ

8 .62

.

4 .a

7 .b

3

B

157


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