81

TRIGONOMETRY.

TRIGONOMETRICAL FORMULE.

1. Express in degrees and in grades, two angles, whose sum is 75 degrees, and difference 25 grades.

2. Find the ratios of the lengths of the decimal and the sexagesimal degree, minute, and second, respectively.

3. Find the circular measure of the angles 57°, 57° 30′, 18°, 54°, and 22° 30'.

4. Find the angles whose circular measures are 1,

I 3 Ι

2

2

Adapt to radius r, the following formula:

10. I+cos 2A.cos 2B = cos2 (A+B) + cos2 (A−B).

II. If I be the length of an arc of 45° to radius r, l' the length

of an arc of 30° to radius r', show that

12.

Prove the following formula:-
cos B-cos A

cos B+cos A

21

=tan (A+B). tan (A—B).

13. cot2 A x cos2 A= cot2 A- cos2 A.

33. sin A+ sin 2A+ sin 3A=sin 2A {3—4 sin24

2

34. 1+ cos A+ cos 2A+ cos 3A=2 cos A { 2 cos2A+cos A−1}. 35. {cos A+ sin A√—1—1}{cos A—sinA√—1—1}=4sin2

A
2

36. {2 sin A+ sin 2A}tan' =2 sin A-sin 2A.

A

40. I+cos 2A. cos 2B = cos2 (A+B) + cos2 (AB). sec A. sec B. cosec A. cosec B

41. sec (A+B)

42. vers (A—B). vers {π— (A+B) } = sin A—sin B)2.

B

43. cos A+cos(A+2 B)=2(2 cos2-1). cos (A+B).

2

44. sin (A+B). sin 3 (A-B)=sin2 (2 A-B)-sin2 (2 B-A). 45. cos (A-B-C) + cos (A-B+C) + cos (A + B-C) +cos (A+B+C)=4 cos A cos B cos C.

46. sin (A-B). sin C-sin (A-C). sin B+ sin (B-C). sin A=0. 47. sin (A+B). sin (B+C+D)

=sin A. sin (C + D) + sin B. sin (A+B+C+D).

48. cos A+ cos (A +2 B) = =2 cos B. cos (A+B).

49. If S and s be the sines of two arcs, C and c their cosines, (S2 — S2s2) (C2 — C2c2) = (s2 — s2S2) (c2 — c2C2).

50. If T and t be the tangents, S and s the secants of two arcs, S'(-1) (T2 + 1)

=

s2 (S2 — 1) ̄ ̄T2 (t2 + 1)

51. Find the values of the sines of the following angles,
7° 30', 9°, and 22° 30′.
52. Find the cosines of the following angles,
12°, 11° 15', and 33° 45'.
53. Find the tangents of the following angles,
18°, 22° 30′, 37° 30′.

54. Find the cotangents of the following angles,
36°, 75°, 225°.

55. Find the secants of the following angles, 22° 30′, 54°, 225°.

56. Find the cosecants of the following angles, 30°, 72, 120°.

57. Find the versines of the following angles, 15°, 67° 30', 240°.

58. Find the chords of the following angles, 36°, 45°, 240°.

Prove the following:

=

59. sin 3A 4 sin A. sin (60°+A). sin (60°-A). 60. 2 sec 2A=sec (45°+A). sec (45°-A).

Prove the following formula, where A+B+C=90°.

69. tan A. tan B+tan A. tan C+tan B. tan C=1.

70. cot A+ cot B+cot C=cot A. cot B. cot C.

71. tan A+tan B+tan C=tan A.tan B.tan C+ sec A. sec B.sec C. 72. sin 2A + sin2B+ sin 2C=4 cos A. cos B. cos C.

Prove the following formula, where A+B+C=180°.

73. sin 2A + sin 2B+ sin 2C-4 sin A. sin B. sin C.

C
+sin2 +2 sin

2

75. cos2A+cos2 B+ cos2 C+2 cos A. cos B. cos C=I.

77. sin (A+B). sin (B+C)=sin A. sin C.

78. tan A+tan B+tan C=tan A. tan B. tan C. 79. cot A. cot B+cot A. cot C+cot B. cot C=1. 80. If A+B+C=45°, prove that

tan A+tan B+tan C-tan A. tan B. tan C

I-tan A. tan B. -tan A. tan C-tan. tan C.

If A, B and C be in arithmetical progression, prove the following formulæ :

81. sin A-sin C=2 sin (A-B). cos B.

82.*(sin A—sin C). sin B= (cos C—cos A). cos B.

84. If cos (A-C). cos B=cos (A-B+C), prove that the tangents of A, B and C are in harmonical progression.

tan B
a sin B
b+a cos B
I + m

tan A.

87. If sin B=m sin (2A+B), then tan (A+B):

=

I-m

A+B A-B

2

88. If cos A=cos B. cos C, then tan
89. If sec A. sec B+tan A. tan B=sec C,
sin C sin A

then tan B±

Determine A from the following equations:

90. sin A sin 2A.

92. tan A=cosec 2A.

94. tan A+ 3cot A=4.

96. cos nA+cos (n-2)A=cos A.
97. tan1A+4 sin2 A=6.
99. tan A+cot A=4.
IOI. m=sin A.cos A.

103. 4 sin A.sin 3A=1.

104.

91. tan 2A=3 tan A.
93. cos A=tan A.
95. 2 sin2 3A+ sin26A=2.

98. 2 sin A=tan A.

100. m sin A:

=n cos2 A.

102. cos 3A+cos 2A+cos A=0.

Determine a from the following equations:-
sin(x+a)+cos(x+a)=sin(x− a) + cos(x—α).

106. tan a. tan x.=tan2 (a+x) — tan2 (a—x').

107. {√1 + sin x−1}.{ √1 — sin x + 1} = tan