Find the general solutions of the following equations :
sin x = tan x
Ideas required to solve the problem:
The general solution of any trigonometric equation is given as –
• sin x = sin y, implies x = nπ + (– 1)n y, 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,
⇒
⇒
⇒
either,
sin x = 0 or cos x = 1
⇒ sin x = sin 0 or cos x = cos 0
We know that,
If sin x = sin y, implies x = nπ + (– 1)n y, where n ∈ Z
∵ sin x = sin 0
∴ y = 0
And hence,
x = nπ where n ϵ Z
Also,
If cos x = cos y, implies x = 2mπ ±y, where m ∈ Z
∵ cos x = cos 0
∴ y = 0
Hence, x is given by
x = 2mπ where m ϵ Z
∴ x = nπ or 2mπ ,where m,n ϵ Z …ans