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Trigonometric Equations – Formulas and Solutions


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 principal  of  trigonometric equation.

 Common Solutions for Trigonometric Equations

Trigonometrical equation Solutions
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

Solution:

⇒ 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.

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