Ex 3.4, 9 - Chapter 3 Class 11 Trigonometric Functions

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Ex 3.4, 9 Find the general solution of the equation sin x + sin3x + sin5x = 0 sin x + sin 3x + sin 5x = 0 (sin x + sin 5x) + sin 3x =0 (sin x + sin 5x) + sin 3x = 0 2 sin (( +5 )/2) . cos (( 5 )/2) + sin 3x = 0 2 sin (6 /2) . cos (( 4 )/2) + sin 3x = 0 2 sin (3x) . cos (-2x) + sin 3x = 0 2 sin 3x . cos 2x + sin 3x = 0 sin 3x (2cos 2x + 1) = 0 Hence sin 3x = 0 or 2cos 2x + 1 = 0 sin 3x = 0 or 2cos 2x = 1 sin 3x = 0 or cos 2x = ( 1)/2 We need to find general solution both separately General solution for sin 3x = 0 Let sin x = sin y sin 3x = sin 3y Given sin 3x = 0 From (1) and (2) sin 3y = 0 sin 3y = sin (0) 3y = 0 y = 0 General solution for sin 3x = sin 3y is 3x = n (-1)n 3y where n Z Put y = 0 3x = n (-1)n 0 3x = n x = /3 where n Z General solution for cos 2x = ( )/ Let cos x = cos y cos 2x = cos 2y Given cos 2x = ( 1)/2 From (3) and (4) cos 2y = ( 1)/2 cos (2y) = cos (2 /3) 2y = 2 /3 General solution for cos 2x = cos 2y is 2x = 2n 2y where n Z putting 2y = 2 /3 2x = n 2 /3 x = 1/2 (2n 2 /3) x =2 /2 1/2 2 /3 x = n /3 where n Z Hence General Solution is For sin3x = 0, x = /3 and for cos 2x = ( 1)/2 , x = n /3 where n Z