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JEE Mains Maths · Formula sheet

Trigonometric Equations formulas

8 formulas and 8 common traps for JEE Mains Maths Trigonometric Equations, grouped by subtopic.

Full notes with worked examples

Quadratic in One Ratio

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Solve for the ratio, then count per period

General solutions

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

Two equations, domain limits and a sized interval

Roots of cos θ = c in order from 0

α, 2π−α, 2π+α, 4π−α, …(α=cos⁡−1c∈(0,π))\alpha,\ 2\pi-\alpha,\ 2\pi+\alpha,\ 4\pi-\alpha,\ \dots\qquad(\alpha=\cos^{-1}c\in(0,\pi))

Common traps

Values ±1 and the endpoints

cos⁡θ=±1\cos\theta=\pm1 or sin⁡θ=±1\sin\theta=\pm1 is reached once per period, not twice. And a closed interval can hold a solution at each end: cos⁡θ=−1\cos\theta=-1 on [−π,π][-\pi,\pi] holds at both −π-\pi and π\pi.

Do not cancel a ratio

In sin⁡θcos⁡θ=sin⁡θ\sin\theta\cos\theta=\sin\theta, cancelling sin⁡θ\sin\theta leaves cos⁡θ=1\cos\theta=1 and loses θ=π\theta=\pi. Write sin⁡θ(cos⁡θ−1)=0\sin\theta(\cos\theta-1)=0 and keep both factors.

Product-to-Sum and Multiple Angles

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Reduce to cos A = cos B

Two families

cos⁡A=cos⁡B⇒A=2nπ+B or A=2nπ−B\cos A=\cos B\Rightarrow A=2n\pi+B\ \text{or}\ A=2n\pi-B

Collapse to one multiple angle

Triple angle

cos⁡3x=4cos⁡3x−3cos⁡x,tan⁡3x=3tan⁡x−tan⁡3x1−3tan⁡2x\cos3x=4\cos^3x-3\cos x,\qquad \tan3x=\frac{3\tan x-\tan^3x}{1-3\tan^2x}

Common traps

Subtract the common angles

The two families overlap wherever both formulas give the same angle, and 00 is always one of them. Adding the two counts without removing the overlap overcounts.

Check the original tangents

Collapsing to tan⁡3x\tan3x or sin⁡4x\sin4x widens the domain. A root of the new equation can make an original tan⁡\tan or sec⁡\sec undefined; substitute each root back and drop those.

Range and Existence of Solutions

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The range decides whether a solution exists

Existence

f(x)=k has a solution exactly when min⁡f≤k≤max⁡ff(x)=k\ \text{has a solution exactly when}\ \min f\le k\le\max f

a cos x + b sin x = c by the auxiliary angle

Auxiliary angle

acos⁡x+bsin⁡x=a2+b2 cos⁡(x−φ),tan⁡φ=baa\cos x+b\sin x=\sqrt{a^2+b^2}\,\cos(x-\varphi),\qquad\tan\varphi=\frac ba

Common traps

The vertex may be outside

The range of at2+bt+cat^2+bt+c on [−1,1][-1,1] does not always reach the vertex value. If the vertex lies outside [−1,1][-1,1], the extremes are g(−1)g(-1) and g(1)g(1).

Squaring adds roots

Squaring acos⁡x=c−bsin⁡xa\cos x=c-b\sin x to get a quadratic in sin⁡x\sin x also brings in the roots of acos⁡x=−(c−bsin⁡x)a\cos x=-(c-b\sin x). Check each root in the original equation, or use the auxiliary angle, which needs no check.

Exponential, Bounded and Graphical Equations

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Powers with sin² and cos² in the exponent

Substitution

t=asin⁡2x:asin⁡2x+acos⁡2x=t+att=a^{\sin^2x}:\qquad a^{\sin^2x}+a^{\cos^2x}=t+\frac at

Bounds, graphs and moduli

Forced equality

f(x)≤m≤g(x) for all x:f(x)=g(x) exactly when f(x)=g(x)=mf(x)\le m\le g(x)\ \text{for all}\ x:\quad f(x)=g(x)\ \text{exactly when}\ f(x)=g(x)=m

Common traps

A root in t may be out of range

For a>1a>1, t=asin⁡2xt=a^{\sin^2x} lies in [1,a][1,a]. A root of the quadratic outside that range gives no angle, just as sin⁡x=2\sin x=2 gives none.

Check the part-branches at the ends

A full branch of tan⁡x\tan x always meets a decreasing line once. A part-branch at the edge of the interval may or may not: compare the line's values there with the range that tan⁡x\tan x covers on that piece.

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