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Trigonometric Equations formula sheet

Every formula in the Trigonometric Equations chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 15 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 formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Principal and general solutions

Principal solutions
7× asked, last 2025
Q I: α,Q II: π−αQ III: π+α,Q IV: 2π−α\begin{aligned}&\text{Q I: }\alpha,\qquad \text{Q II: }\pi-\alpha\\&\text{Q III: }\pi+\alpha,\qquad \text{Q IV: }2\pi-\alpha\end{aligned}
Roots in [0,2π)[0,2\pi); α\alpha acute with the same absolute value; keep quadrants with the right sign.
sin and cos
13× asked, last 2026
sin⁡θ=sin⁡α⇒θ=nπ+(−1)nαcos⁡θ=cos⁡α⇒θ=2nπ±α\begin{aligned}&\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^n\alpha\\&\cos\theta=\cos\alpha\Rightarrow\theta=2n\pi\pm\alpha\end{aligned}
n∈Zn\in\mathbb Z. Both together: θ=2nπ+α\theta=2n\pi+\alpha. For 3x3x with x∈[a,b]x\in[a,b], use 3x∈[3a,3b]3x\in[3a,3b].
tan and squares
16× asked, last 2026
tan⁡θ=tan⁡α⇒θ=nπ+αsin⁡2θ=sin⁡2α⇒θ=nπ±α\begin{aligned}&\tan\theta=\tan\alpha\Rightarrow\theta=n\pi+\alpha\\&\sin^2\theta=\sin^2\alpha\Rightarrow\theta=n\pi\pm\alpha\end{aligned}
Square form also for cos⁡2\cos^2, tan⁡2\tan^2. tan needs θ≠(2n+1)π/2\theta\ne(2n+1)\pi/2.
Special values
sin⁡θ=0, tan⁡θ=0: nπ;cos⁡θ=0: (2n+1)π2sin⁡θ=1: 2nπ+π2;cos⁡θ=1: 2nπcos⁡θ=−1: (2n+1)π\begin{aligned}&\sin\theta=0,\ \tan\theta=0:\ n\pi;\qquad \cos\theta=0:\ (2n+1)\tfrac\pi2\\&\sin\theta=1:\ 2n\pi+\tfrac\pi2;\qquad \cos\theta=1:\ 2n\pi\\&\cos\theta=-1:\ (2n+1)\pi\end{aligned}
∣k∣<1|k|<1: sin⁡θ=k\sin\theta=k and cos⁡θ=k\cos\theta=k have two roots in every 2π2\pi.

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Practise Trigonometric Equations

11 more formulas 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 is solvable5×

a cos x + b sin x = c and substitutions

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

Factorising, compound angles and domain

  • Factorising6×
  • Spotting a compound angle5×
  • Domain and lost roots9×

Counting and summing solutions

  • Counting roots18×
  • 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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Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

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