The number of solutions of sin ⁡ x + sin ⁡ 3 x + sin ⁡ 5 x = 0 in the interval [ π 2 , 3 π 2 ] is

Step-by-step solution

sin x + sin 5x + sin 3x = 0 2 sin 3x cos 2x + sin 3x = 0 sin 3x 2 cos 2x + 1 = 0 sin 3x = 0 and 2 cos 2x + 1 = 0 sin ⁡ 3 x = 0 Graph f(x) = sin 3x x = 2 π 3 or x = π or x = 4 π 3 , ∴ sin 3 x = 0 have 3 solutions Graph f(x) = cos 2x cos 2 x = - 1 2 ∴ x = 2 π 3 a n d 4 π 3 So the solution of given equation in [ π 2 , 3 π 2 ] are 2 π 3 a n d 4 π 3