# PRINCIPAL SOLUTION AND GENERAL SOLUTION OF TRIGONOMETRIC FUNCTIONS

## About "Principal Solution and General Solution of Trigonometric Functions"

Principal Solution and General Solution of Trigonometric Functions :

Here we are going to see how to find principal solution and general solution of trigonometric functions.

Principal Solution

The smallest numerical value of unknown angle satisfying the equation in the interval [0, 2π] (or) [−π, π] is called a principal solution.

Principal value of sine function lies in the interval

[−π/2, π/2]

Principal value of cosine function is in

[0, π]

Principal value of tangent function is in

(-π/2, π/2)

General Solution :

The solution of a trigonometric equation giving all the admissible values obtained with the help of periodicity of a trigonometric function is called the general solution of the equation.

 Trigonometric equationsin θ = 0cos θ = 0tan θ = 0sin θ = sinα, where α ∈ [−π/2, π/2]cos θ = cos α, where α ∈ [0,π]tan θ = tanα, where α ∈ (−π/2, π/2) General solutionθ = nπ; n ∈ Zθ = (2n + 1) π/2; n ∈ Zθ = nπ; n ∈ Zθ = nπ + (−1)n α, n ∈ Zθ = 2nπ ± α, n ∈ Zθ = nπ + α, n ∈ Z

## Principal Solution and General Solution of Trigonometric Functions - Examples

Question 1 :

Find the principal solution and general solutions of the following:

(i)  sin θ = −1/√2

Solution :

sin θ = −1/√2

θ = sin -1(−1/√2)

Principal solution :

We have to choose the principal solution between the interval [−π/2, π/2] .

θ  =  (-π/4)

General solution :

θ = nπ + (−1)n α, n ∈ Z

θ = nπ + (−1)n (-π/4), n ∈ Z

(ii) cot θ = √3

Solution :

cot θ = √3

1/tan θ = 1/√3

Taking reciprocals on both sides

tan θ = √3

Principal solution :

We have to choose the principal solution between the interval (−π/2, π/2)

θ  =  π/6

General solution :

θ = nπ + α, n ∈ Z

θ = nπ + (π/6)

(iii) tan θ = -1/√3

Solution :

tan θ = -1/√3

Principal solution :

We have to choose the principal solution between the interval (−π/2, π/2)

θ  =  -π/6

General solution :

θ = nπ + α, n ∈ Z

θ = nπ + (-π/6)

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