Let f x = 4 x + 1 - cos ⁡ x - sin ⁡ x 6 8 sin ⁡ α 0 12 sin ⁡ α 16 s i n 2 α 1 + 4 sin ⁡ α and f 0 = 0 . If the sum of all possible values of α is k π for α ∈ 0,2 π , then the value of k is equal to

Step-by-step solution

f 0 = 1 - 1 0 6 8 sin ⁡ α 0 12 sin ⁡ α 16 s i n 2 α 1 + 4 sin ⁡ α = 0 ⇒ 1 + 4 sin ⁡ α 6 + 8 sin ⁡ α = 0 ⇒ sin ⁡ α = - 1 4 , - 3 4 Let, sin ⁡ a = - 1 4 , sin ⁡ b = - 3 4 ( ∵ both values are negative so the solution lies in 3 r d or 4 t h quadrant) sin ⁡ α = sin ⁡ a , sin ⁡ α = sin ⁡ b α = π + a , 2 π - a , π + b , 2 π - b Hence, sum of all possible values = 6 π