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24. If sin ( = a, and tan 0 =b, prove that

(1 – a?) (1 +62)=1.

Express the following functions in terms of the functions of acute angles less than 45°:

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35. tan 1020o, sec 1395o, sin 1485°.

-V3, V2,

1

36. sin (-240°), cot (-675°), cosec (- 690°).

V3
Ans.

2 37. cos (-300°), cot (-315o), cosec (-1740°).

1

1, 2.

2

1,

2 V3 1

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Find the value of the sine, cosine, and tangent of the following angles :

V3 1 39. – 300°.

Ans.

V3. 2 ? 2 1

1 40. - 135o.

1. V2 1

1 41. 750°.

2'

V3

/3

V3 1 42. - 840°.

2

V3.

V3 1 43. 1020°.

2 2'

-V3.

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Prove, drawing a separate figure in each case, that
46. sin 340°= sin(- 160°).
47. sin (-40°)=sin 220°.
48. cos 320°= -cos (140°).
49. cos (-380°)=- cos 560°.
50. cos 195° :

=- cos (– 15°).

51. cos 380o=

cos 560°.

O

66

66

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52. cos (– 225°)=- cos (– 45°). 53. cos 1005°=

cos 1185°.

1 54. Draw an angle whose sine is

2 55.

cosecant is 2. 56.

tangent is 2. 57. Can an angle be drawn whose tangent is 427 ? 58.

cosine is?

4 59.

secant is 7 ? 60. Find four angles between zero and + 8 right angles which satisfy the equations

1 (1) sin A = sin 20°, (2) sin A=

(3) sin A

7 Ans. (1) 20°, 160°, 380, 520°; 57 77 13 15 T

7 (2)

; 4' 4' 4 4 87 137 22 277

л

27π )

7' 7 7 7 61. State the sign of the sine, cosine, and tangent of each of the following angles :

(1) 275°; (2) – 91°; (3) 193o; (4) — 350°;
(5) – 1000°; (6) 2n+ +

4
Ans. (1) -, +, -; (2) -, -, +; (3) +, -, - ;

(4) +, +, +; (5) +, +, +; (6) +, -, Prove the following identities : 62. (sin’6 + cos26)=1. 63. (sin'0 - cos)'=1-4cos 0 + 4 cos+ 0. 64. (sin 0 + cos()=1+2 sin cos 6. 65. (sec -tan) (sec 0 + tan ) = 1.

T

п

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3)

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66. (cosec 0 — coto) (cosec 0 + cot0)=1. 67. sino 0 + cos30 =(sin 0 + cos 0) (1 – sin 0 cos ). 68. sin A + cosa sino 0 + cos4 A – sin’O cos0. 69. sino0 tan’0 + cos” O cot20 tano 0 + cot20 1. 70. sino tan’0+ coseco secʻ0= 2 tan 0 seco— cosec 0+ sin 0. 71. cos 0 — sin’8 = (cos 0 — sin 0) (1 + sin 0 cos 6). 72. sin 0 + cos o 1-3 sino cos20. 73. tana + tanß= tan o tanß (cota + cot B). 74. cota + tan ß = cot a tan ß (tan « + cotß). 75. 1 – sind=(1 + sin o) (sec — tana). 76. 1+ cosa=(1 – cosa) (coseca coto). 77. (1+ sina + cosa)?=2(1 + sin o) (1 + cosa). 78. (1 – sin a — cos a)'(1 + sin « + cosc)?= 4 sin’a cos?a. 79. 2 vers a

vers’ (= sin’d. 80. verso (1 + cosa)= sin’o.

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