Class 11 Maths Trigonometric Functions Principal and General Solution

Principal and General Solution

 Principal solutions: The solutions where 0 ≤ x < 2π.

 General solution: The expression involving integer ‘n’ this gives all solutions of a trigonometric equation.   To derive general solution we will use the fact that:

  • Values of sinx  repeat after an interval of 2π
  • Values of cos x repeat after an interval of 2π
  • Values of tanx repeat after an interval of π.

Theorem 1: For any real numbers x and y, sin x = sin y implies x = nπ + (–1)n y, where n ∈ Z

Theorem 2: For any real numbers x and y, cos x = cos y, implies x = 2nπ ± y, where n ∈ Z

Theorem 3: If x and y are not odd mulitple of π/2 , then tan x = tan y implies x = nπ + y, where n ∈ Z

 

Refer ExamFear video lessons for Proofs.

 

Numerical: Find the principal & general solutions of the equation sin x = ½ 

Solution:  Given, sin x = ½ 

We know that Sin 30o = ½

Thus x = 30o   = π/6

Using theorem, For any real numbers x and y, sin x = sin y implies x = nπ + (–1)n y, where n ∈ Z

Generic solution is  x = nπ + (–1)n π/6  , where n ∈ Z

 

For more Numericals, refer examfear video lessons.

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