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Trigonometric Equations revision notes

Every idea in the Trigonometric Equations chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 14 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Principal and general solutions

Principal solutions
7× asked, last 2025
Q I: α,Q II: π−α,Q III: π+α,Q IV: 2π−α\text{Q I: }\alpha,\qquad \text{Q II: }\pi-\alpha,\qquad \text{Q III: }\pi+\alpha,\qquad \text{Q IV: }2\pi-\alpha
Roots in [0,2π)[0,2\pi). α\alpha = acute angle with the same absolute value; keep the quadrants where the ratio has the right sign.
α\alphaπ−α\pi-\alphaπ+α\pi+\alpha2π−α2\pi-\alphaα\alphaα\alphaα\alphaα\alpha
The same acute α\alpha in every quadrant; keep the quadrants where the ratio has the sign you need
General solution: sin and cos
13× asked, last 2026
sin⁡θ=sin⁡α⇒θ=nπ+(−1)nα,cos⁡θ=cos⁡α⇒θ=2nπ±α\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^n\alpha,\qquad \cos\theta=\cos\alpha\Rightarrow\theta=2n\pi\pm\alpha
n∈Zn\in\mathbb Z. Usually α∈[−π/2,π/2]\alpha\in[-\pi/2,\pi/2] for sin, α∈[0,π]\alpha\in[0,\pi] for cos. So cos⁡A=cos⁡B\cos A=\cos B means A=2nπ±BA=2n\pi\pm B.
xxyy−π-\piπ\pi2π2\pi3π3\pi−π−α-\pi-\alphaα\alphaπ−α\pi-\alpha2π+α2\pi+\alpha3π−α3\pi-\alpha
sin⁡θ=12\sin\theta=\frac12: roots nπ+(−1)nαn\pi+(-1)^n\alpha, alternately α\alpha after and α\alpha before each multiple of π\pi
xxyyOπ3\frac\pi32π3\frac{2\pi}3π\picos⁡3x=12\cos3x=\frac12
x∈[0,π]x\in[0,\pi] means 3x∈[0,3π]3x\in[0,3\pi]: three roots
  • sin⁡θ=sin⁡α\sin\theta=\sin\alpha and cos⁡θ=cos⁡α\cos\theta=\cos\alpha together: θ=2nπ+α\theta=2n\pi+\alpha (the same angle, one full turn at a time).
  • Not one root per period. For ∣k∣<1|k|<1, sin⁡θ=k\sin\theta=k and cos⁡θ=k\cos\theta=k have two roots in any half-open 2π2\pi interval, e.g. [0,2π)[0,2\pi): α,π−α\alpha,\pi-\alpha (sin), α,2π−α\alpha,2\pi-\alpha (cos), mod 2π2\pi. Only k=±1k=\pm1 gives one.
  • Not [0,2π][0,2\pi] for 3x3x. If x∈[a,b]x\in[a,b] then 3x∈[3a,3b]3x\in[3a,3b]. cos⁡3x=1/2\cos3x=1/2 with x∈[0,π]x\in[0,\pi]: 3x=π/3, 5π/3, 7π/33x=\pi/3,\,5\pi/3,\,7\pi/3, so three roots, not two.
General solution: tan and squares
16× asked, last 2026
tan⁡θ=tan⁡α⇒θ=nπ+α,sin⁡2θ=sin⁡2α⇒θ=nπ±α\tan\theta=\tan\alpha\Rightarrow\theta=n\pi+\alpha,\qquad \sin^2\theta=\sin^2\alpha\Rightarrow\theta=n\pi\pm\alpha
The square form also holds for cos⁡2\cos^2 and tan⁡2\tan^2 and covers both signs at once. tan⁡\tan needs θ≠(2n+1)π/2\theta\ne(2n+1)\pi/2.
xxyy−π-\piπ\piπ4\frac\pi4y=1y=1
tan⁡θ=1\tan\theta=1: one root per branch, π\pi apart: θ=nπ+π4\theta=n\pi+\frac\pi4
  • Special values: sin⁡θ=0\sin\theta=0 or tan⁡θ=0\tan\theta=0 gives nπn\pi; cos⁡θ=0\cos\theta=0 gives (2n+1)π/2(2n+1)\pi/2; sin⁡θ=1\sin\theta=1 gives 2nπ+π/22n\pi+\pi/2; cos⁡θ=1\cos\theta=1 gives 2nπ2n\pi; cos⁡θ=−1\cos\theta=-1 gives (2n+1)π(2n+1)\pi.

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Practise Trigonometric Equations

11 more concepts in 4 topics

Equations reducible to a quadratic; a cos x + b sin x = c and substitutions; Factorising, compound angles and domain; and Counting and summing solutions. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Equations reducible to a quadratic

  • Quadratic in one ratio16×
  • Exponential forms4×
  • When f(x) = p has a solution5×

a cos x + b sin x = c and substitutions

  • a cos x + b sin x = c2×
  • Substitution t = sin x + cos x3×

Factorising, compound angles and domain

  • Factorising by sum-to-product6×
  • Spotting a compound angle5×
  • Domain and lost roots9×

Counting and summing solutions

  • Counting roots in an interval18×
  • Sum of solutions12×
  • Graphical method5×

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About 50 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Trigonometric Equations

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