Number of roots of the equation cos 2 ⁡ x + 3 + 1 2 sin ⁡ x - 3 4 - 1 = 0 which lie in the interval - π , π is

Step-by-step solution

Given equation is 1 - sin 2 ⁡ x + 3 + 1 2 sin ⁡ x - 3 4 - 1 = 0 ⇒ sin 2 ⁡ x - 3 + 1 2 sin ⁡ x + 3 4 = 0 ; 4 sin 2 ⁡ x - 2 3 sin ⁡ x - 2 sin ⁡ x + 3 = 0 2 sin x ( 2 sin x − 3 ) − ( 2 sin x − 3 ) = 0 ⇒ ( 2 sin x − 1 ) ( 2 sin x − 3 ) = 0 On solving we get sin ⁡ x = 1 2 ; 3 2 x = π 6 , 5 π 6 ; π 3 , 2 π 3