General Solutions of Trig EquationsJEE Main

All the x's that satisfy sin x = k — interactive Mathematics simulation for IIT-JEE.

Concept

Trig functions repeat forever, so equations like sin x = ½ have infinitely many solutions. The general-solution formulas package them all with one integer n: memorise the trio — sin: $n\pi + (-1)^n\alpha$ ; cos: $2n\pi \pm \alpha$ ; tan: $n\pi + \alpha$ — where α is the principal value.

Key formula

\sin x = \sin\alpha: x = n\pi + (-1)^n\alpha; \quad \cos x = \cos\alpha: x = 2n\pi \pm \alpha; \quad \tan x = \tan\alpha: x = n\pi + \alpha

Derivation

sin: solutions in one period are α and π − α; stitching periods gives the (−1)ⁿ zigzag.

cos is even → symmetric pair ±α, repeating every 2π. tan has period π itself → single arithmetic progression. Special cases worth caching: sin x = 0 → x = nπ; cos x = 0 → x = (2n+1)π/2; sin x = 1 → x = 2nπ + π/2.

Scenarios to explore

General Solutions of Trig Equations — Every x with sin x = k — the nπ + (−1)ⁿ family visualised.

Real-world applications

Finding ALL times a wave hits a level (signals, tides, AC zero-crossings).
Intersection points of trig graphs.
Eigen-frequency conditions in standing-wave problems.

JEE exam tips

sin θ = sin α ⟺ θ = nπ + (−1)ⁿα — also usable to EQUATE angles inside equations.
Squaring introduces extras: solve sin x = cos x as tan x = 1, not by squaring.
Count solutions in an interval: solve generally, then bound n.

Common mistakes

Reporting only the principal value when 'all solutions' is asked.
Using the sin formula for cos equations.
|k| > 1 for sin/cos — no solutions, not 'calculator error'.

Exam traps to avoid

sin²x = sin²α ⟺ x = nπ ± α (its own cleaner formula).
Don't divide by cos x mid-equation — you lose the cos x = 0 family; factor instead.