EXAMPLES.

1.

PROVE that 45°, 15′, 20′ = 508, 28', 39", 50"; 11°, 15′, 37′′

180°

= 11, 40', 3", 9""; 18°, 12′ = 20, 20';

= 115, 47';

√3

2.

=

The Complements of 17°, 36', 43"; 29", 27', 6"-32; and 216, 45'; are 72°, 23', 17"; 60°, 32', 53"-68; and

126, 45'.

3. The Supplements of 37°, 4, 3"; 115°, 13′, 24" 66; and 226°, 14′, 17′′; are 142°, 55'′, 57′′; 64°, 46′, 35′′·34; and - (46°, 14′, 17′′).

3

4. If Cot A find the values of sin A, cos A, cosec A,

= - "

versin A, and sec A.

5. Prove the formulæ,

(1). (Sec A. cosec 4) = (sec A)2 + (cosec A)".

(2). (cot A. cos 4)2= (cot A)2- (cos A)".

6. (1). If (tan A)2 + 4 (sin A)2 = 6; A = 60o.

(2). If m = tan A + sin A, and n = tan A − sin A; Cos A:

m- n

m + n

=

(9). Cot 24. cosec 2 A cot A. (cosec A) - tan A. (sec )2. (10). Versin (180° – A) = 2 vers 1⁄2 (180o + A) . vers 1⁄2 (180o — A).

1242

(1). Cos 2 A + cos 2 B = 2 . cos (A + B) . cos (A – B).

(2). 2. {(sin A. sin B)2 + (cos A. cos B)} = 1 + cos 2 A. cos 2 B.

(Cos A + B)2 - (sin A)2 ·
= cos B. cos (2 A + B).

(3).

Sin (A – B)

(4).

(5).

9.

= 0.

Cos (A + B). sin (A – B)+cos (B + C) sin (B − C)

+ cos (C+D). sin (C-D)+cos (D+A). sin (D-A)=0.

Prove that

(2).

Sec (45° + A). sec (45o – A) = 2 sec 2 A.

(3). 2 sec Atan (45° + A) + cot (45o + 1⁄2 4).

10. Find the numerical values of Sin 15°, Sin 9°, Versin 15°, Tan 22°, 12', Sin 150°, Cos 135°, Sin 3o, 22o,12′, Sec 225°; and prove that

Cos 12o,
Sin 75°,

(1).

Tan 50°+ cot 50° = 2 sec 10°.

(2).

A
2 cos =
2n

2 + √ {2 + &c + √ (2 + 2 cos ▲) } ;

is repeated n times,

where n is an integer, and the symbol

-each time affecting all the quantities which follow it.

(9).

(2).

(3).

(4).

Tan A + cot 2 A = sin A . (1 + tan A . tan 1⁄2 4).

(10.) Cos n A+ cos (n − 2) A = cos A.

12. Determine in the following equations :

(1). Sin (ax) = cos (a + x).

Sin (x + a) + cos (x + a) = sin (x − a) + cos (x − a).
Sina+sin(x - a) + sin (2 x + a) = sin(x+a) + sin (2 x − a).
Tana. tan x = (tan a + x)2 - (tan a -x)2;—to find cos x.

(8).

n. (sec x)°. tan (a − x) = m. (sec a − x)2. tan x.

(1). Tan A.tan B + tan A. tan C + tan B. tan C = 1.

(2). Cot A+ cot B + cot C

=

cot A. cot B. cot C.

(3). Tan A+tan B+tan C=tan A. tan B.tan C+ sec A. sec B. sec C.

14. If A + B + C = 180°;

(1). 4 sin A. sin B. sin C = sin 2A + sin 2 B + sin 2 C.

(2). (Sin4)2+(sin B)2+ (sin C)2+2 sin 4. sin B. sin C=1.

16. (1). If m = tan sin 0, and find the relation between m and n.

(2). If (x − a). cos d + c. sin (1 − d)

= 0

(y-b). cos d+c. cos (1-8)=
d) 0;

n = tan + sin ;

then (x − a). sin l + (y − b) cos l + c = 0.

(3).

(4).

If a. (sin 0)2+ a'. (cos 0)2 = b, a'. (sin (')2 + a. (cos 0')2=b';

(5).

If Cos v =

m2 - 1

; then Tan N

v =