Equations which include the trigonometric functions of unknown angle are called trigonometric equations.
For example, sin x = 1/2, cos x = sin 2x, etc.
Solution of a Trigonometric Equations
A solution of a solution of trigonometric equations to find the value of unknown angle which satisfies the equation. The solution of a trigonometric equation(sin x = 0) for which x = 0, π, 2π. This is called principle of trigonometric equation.
Common Solutions for Trigonometric Equations
|sin x = 0||x = nπ|
|cos x = 0||x = (nπ + π/2)|
|tan x = 0||x = nπ|
|sin x = 1||x = (2nπ + π/2) = (4n+1)π/2|
|cos x = 1||x = 2nπ|
|sin x = sin θ||x = nπ + (-1)nθ, where θ ∈ [-π/2, π/2]|
|cos x = cos θ||x = 2nπ ± θ, where θ ∈ (0, π]|
|tan x = tan θ||x = nπ + θ, where θ ∈ (-π/2 , π/2]|
|sin2 x = sin2 θ||x = nπ ± θ|
|cos2 x = cos2 θ||x = nπ ± θ|
|tan2 x = tan2 θ||x = nπ ± θ|
Trigonometric Equations Examples
Let us see some an example to have a better understanding of trigonometric equations, which is given below:
Example 1: Find the general solution of sin 3x =0
⇒ Sin 3x = 0
⇒ 3x = nπ
⇒ x = nπ/3
Example 2: sin 2x – sin 4x + sin 6x = 0
Solution:Given: sin 2x – sin 4x + sin 6x = 0
⇒sin 2x + sin 6x – sin 4x = 0
⇒2sin 4x.cos 2x – sin 4x = 0
⇒sin 4x (2cos 2x – 1) = 0
⇒sin 4x = 0 or cos 2x = ½
⇒4x = nπ or 2x = 2nπ ± π/3
Therefore, the general solution for the given trigonometric equation is:
⇒x = nπ/4 or nπ ± π/6
Example 3: Find the principal solution of the equation sin x = 1/2.
Solution: Since we know that sin π/6 = 1/2
∴ sin 5π/6
and sin (π – π/6) = sin π/6 = 1/2
Hence, the principal solutions are x =π/6 and x = 5π/6.