(13) If ẞ and y be two values of which satisfy (14) Given a2 cos a cos ẞ+a (sin a + sin ẞ) +1=0, prove that and that a2 cos ẞ cos y+a (sin ẞ+sin y) +1=0, cos a+cos B+cos y=cos (a+B+y), B and y being unequal and less than π. (15) If 1 and 2 are two values of which satisfy shew that 1 and 0, if substituted for 0 and ø in the equation will satisfy it. (16) Solve the equations cos (0+a)=sin & sin ß, cos (+8)=sin sin a, and shew that if 1 and 1⁄2 be the two values of p, n2 cannot be greater than 1+2m cos a+m2. (18) Eliminate 0 and 4 from the equations GENERAL MISCELLANEOUS EXAMPLES. XLIII. N.B. For convenience in printing, some writers use n! to denote 1.2.3...n. (1) A person walks from one end A of a wall a certain distance a towards the West, and observes that the other end B then bears E.S. E. He afterwards walks from the end Ba distance (√2+1) a towards the South, and finds that the end ▲ bears N. W. Shew that the wall makes an angle cot-12 with the East. (2) A man on the top of a hill observes the angles of depression a, ẞ, y of three consecutive milestones on a straight horizontal road running directly towards him; prove that the height of the hill is (3) sin 2 (B+y) + sin 2 (y+a) + sin 2 (a+ß) =4 sin a sin ß sin y cos (a+B+y) +4 cos a cosẞ cos y sin (a+B+y). (4) If 2a+23+2y=nTM, n-1 then sin 2 (B+y)+sin 2 (y+a)+sin 2 (a+B) =2(−1)2 {1-(-1)"} cos a cos ẞ cos y n +2 (−1)3 {1+(−1)"} sin a sin ẞ sin y. (8) Eliminate from the equations (a+b)(x+y)=cos @ (1 + 2 sin2 8), (a-b) (x-y)=sin (1+2 cos2 ). (9) If cos (0-4) is a mean proportional between cos (8+) and sin (+), then cosec 20+ sec 20=cosec 24 + sec 2p. (10) If A+B+C=90°, then cosec A + cosec B + cosec C - 2 =cot B tan C+ cot C tan B+ cot C'tan A+ cot A tan C +cot A tan B+ cot B tan A. (11) If tan (B+C-A) tan (C+A-B) tan (A+B-C)=1, sin 44 sin 4B sin 4C-4 cos 24 cos 2B cos 2C. then (12) If 4 (a+B+y)=π, prove that cos (68+4y-8a) + cos (6y+4a-88)+cos (6a+ 48 - 8y) (13) If = 4 cos (5a - 2ẞ - y) cos (5ẞ - 2y - a) cos (5y – 2a – B). = 1-x2 y2-22-2.xyz, prove by trigonometry, that x √(1 − x2) + y √(1 − y2) + z √√(1 − 22) = 2 √ √{(1 − x2) (1 − y2) (1 − z2}. (14) The formulae (2n+1)π±α, (n − 1) π + ( − 1)" (≥π—a) represent the same series of angles. (15) If A+B+C=(2m+1) π or 2m+, then (sin A+cos A) (sin B+cos B) (sin C+cos C)=2 sin A. sin B. sin C +2 cos A cos B. cos C+1; and if A+B+C=2mñ oг 2mñ – 1⁄2Ã, then (sin A+cos A) (sin B+cos B) (sin C+cos C)=2 sin A sin B sin C +2 cos A cos B cos C-1. (16) If sin (2n+1) A sin (B-C)+sin (2n+1) B sin (C-A) +sin (2n+1) C sin (A-B)=0, where n is an integer, then sin (n-1) A sin (n+1) (B-C)+sin (n-1) B sin (n+1) (C−A) +sin (n-1) Csin (n+1)(A-B)=0. (17) If sin 20+ sin 2p=sin 2a, prove that the three expressions cos e cos (a+) sin 0 sin (a — 0), unequal, then each of these fractions is equal to = sin (B+y) sin (y+a) sin (a+B) cos (B+y) cos (y+a) cos (a+ß)+sin2 (a+ß+y)* and cos 0: (20) If √2 cos A=cos B+cos3 B, √2 sin Asin B-sin3 B, then sin (B-A)= cos 2B=3. (21) If 4 cos(x − y) cos (y − z) cos (z--x)=1, prove that 1+12 cos 2 (x − y) cos 2 (y-z) cos 2 (z− x) (22) If =4 cos 3(x-y) cos 3 (y-z) cos 3 (z - x). sin (B+y) − k sin (a+d)=sin (y+a) — k sin (B+8) where a, ß, y are unequal and each less than π, then will k2=1, and each member of the equations =0. = cos (B+a)+cos(a+y)_ cos (B+y)+cos (a+y) (23) If then and (24) If x2 cos a cos ẞ+x (sin a+ sin ẞ)+1=0, x2 cos ẞ cos y +x (sin ẞ+sin y) +1=0, x2 cos y cos a +x (sin y + sin a) +1=0. prove that prove that x=4 cos (a – B) cos (B − y) cos 1 (y − a). prove that either or must be of the form n+1π. |