76. sec1ğ+sec¬1 13 = 90°.

77. tan-1(2+√3)-tan-1 (2-√3)=sec-12. 78. tan-1+tan-1+tan-11+tan~1}=45°. 79. Given cos x = , find sinx and cos x.

80. Given tan x = , find tan x.

8

81. Given sin x + cos x= √, find cos 2x.

82. Given tan 2x=24, find sin x.

83. Given cos 3x= 23, find tan 2x.
84. Given 2 cscx · cotx=√√3, find sinx.
85. Find sin 18°, cos 36°.
Solve the following equations:
86. sin x 2 sin (+x).

90. sin x cos 2x 4 sin2 x. 91. 4 cos 2x + 3 cos x=1. sin x + sin 2x sin 3x.

92.

93.

99. sinx+sin 2x=1-cos 2x. 100. sec 2x+1=2 cosx.

101. tan 2x+tan 3x=0.

102. tan (+x)+tan (π--x)=4. 103. √1+sina-√1-sinx-2 cos x.

110. cos 3x+8 cos3x=0.

111. sec (x+120°) + sec (x-120°) = 2 cos x.

123. cos 3x-2 cos 2x + cos x=0.

124. tan 2x tan x = 1.

125. sin (x+12°) + sin (x — 8°) = sin 20°. 126. tan (60°+x) tan (60°-x)=-2.

127. sin (x+120°) + sin (x +60°) = 3.

128. sin (x+30°) sin (x — 30°) = 1.

129. sin1x+cosx§.

130. sinx-cos*x = 275•

131. tan (x+30°) = 2 cos x.

132. secx=2 tan x +4.

133. sin (xy) = cos x, cos (x+y)=sinx.

=cosx,

134. tan x+tan y=a; cot x+cot y=b.

135. sin(x+12°) cos (x — 12°) = cos 33° sin 57°.

136. sin-1x + sin1x=120°.

137. tan-1x+tan-12x-tan-13√3.

Solve the following equations:

-1

139. sin 1x+3 cos1x=210°.

140. tan-1x+2 cot¬1x=135°.

141. tan-1(x+1)+ tan-1 (x-1)=tan-12x.

2. Draw the curve of tangents, and show the changes in the value of this function as the arc increases from 0° to 360°.

3. In terms of functions of positive angles less than 45°, express the values of sin-250°, csc 13π, tan — 16 π. Also find all the values of 0 in terms of a when cos 0= √sin2a.

4. (a) Given cos x=0.5, find cos 2x and tan 2 x.

(b) Prove that vers (180° — A) +vers (360° — A) = 2.

5. Prove the check formulæ :

* NOTE.

a+b:c=cos (A-B): sin C;
a-b:c=sin(A — B) : cos † C.

In these papers, as in many text-books, the Greek letters a (alpha), B (bayta), y (gamma), 8 (delta), 0 (thayta), and 4 (phee), are occasionally used to denote angles.

6. In a right triangle, r (the hypotenuse) is given, and one acute angle is n times the other; find the sides about the right angle in terms of r and n.

7. The tower of McGraw Hall is 125 ft. high, and from its summit the angles of depression of the bases of two trees on the campus, which stand on the same level as the Hall, are respectively 57° 44′ and 16° 59', and the angle subtended by the line joining the trees is 99° 30'. Find the distance between the trees.

4. Given tan 45° -1; find all the functions of 22° 30'.

5. Determine the number of solutions of each of the triangles: a=13.4, b=11.46, A=77° 20′; c=58, a=75, C=60°; b=109, a=94, A=92°10'; c=309, b=360, C-21° 14' 25".

6. In a parallelogram, given side a, diagonal d, and the angle A formed by the diagonals; find the other diagonal and the other side.

7. A and B are two objects whose distance, on account of intervening obstacles, cannot be directly measured. At the summit of a hill, whose height above the common horizontal