Let 0 < θ 1 < θ 2 < θ 3 < … denotes the positive solutions of the equation 3 + 3 cos ⁡ θ = 2 sin 2 ⁡ θ . If θ 3 + θ 7 = a π , where a is an integer, then the value of a is equal to

Step-by-step solution

3 + 3 cos ⁡ θ = 2 sin 2 ⁡ θ = 2 - 2 cos 2 ⁡ θ . ⇒ 2 cos 2 ⁡ θ + 3 cos ⁡ θ + 1 = 0 ⇒ cos ⁡ θ = - 1 , - 1 2 ⇒ θ = 2 n + 1 π or 2 n π ± 2 π 3 ⇒ θ = 2 π 3 , π , 4 π 3 , 8 π 3 , 3 π , 10 π 3 , 14 π 3 , 5 π i.e. θ 3 + θ 7 = 4 π 3 + 14 π 3 = 18 π 3 = 6 π ⇒ a = 6