|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:
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.