Find the general solutions of the following equations :
sin 2x + cos x = 0
Ideas required to solve the problem:
The general solution of any trigonometric equation is given as –
• sin x = sin y, implies x = nπ + (– 1)ny, where n ∈ Z.
• cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.
• tan x = tan y, implies x = nπ + y, where n ∈ Z.
Given,
We know that: sin θ = cos (π/2 – θ)
∴
⇒
We know that: -cos θ = cos (π – θ)
∴
⇒
If cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.
From above expression and on comparison with standard equation we have:
y =
∴
Hence,
or
∴ or
⇒ or
∴ ,where n ϵ Z