The Chapter in One Idea

Everything here flows from one construction:

Wrap the real line around the unit circle: cos x and sin x are the coordinates of where you land. Every identity is the geometry of that circle written in algebra.

From this: angle measures (Section 1), values and periodicity (Section 2), signs-domains-graphs (Section 3), the sum-difference machine (Section 4), and multiple angles with product-sum transformations (Section 5).

Exam weight at a glance: JEE Main draws on this chapter every single year — identities, ranges of asin⁡x+bcos⁡xa\sin x + b\cos x, values like sin 18°, and trigonometric equations (JEE Corner). It also silently underwrites complex numbers (polar form), vectors, and all of calculus. If one chapter of Class 11 deserves over-preparation, it is this one.

Measures, Values and Signs — Formula Card

  • Conversions: π\pi rad = 180°; radian = π180×\frac{\pi}{180} \times degree; 1 rad ≈\approx 57°16'. Arc length: l=rθl = r\theta (θ\theta in radians only!).
  • Unit circle: P = (cos x, sin x); cos⁡2x+sin⁡2x=1\cos^2 x + \sin^2 x = 1; companions 1+tan⁡2x=sec⁡2x1 + \tan^2 x = \sec^2 x, 1+cot⁡2x=csc⁡2x1 + \cot^2 x = \csc^2 x.
  • Value row (sin): 0,12,12,32,10, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1 at 0,π6,π4,π3,π20, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}; cos runs it backwards; tan = sin/cos.
  • Zeros: sin x = 0 at nπn\pi; cos x = 0 at (2n+1)π2(2n+1)\frac{\pi}{2}.
  • Even-odd: sin⁡(−x)=−sin⁡x\sin(-x) = -\sin x (odd; also tan, cot, cosec); cos⁡(−x)=cos⁡x\cos(-x) = \cos x (even; also sec).
  • Periods: 2π2\pi for sin, cos, sec, cosec; π\pi for tan, cot.
  • ASTC: All (I), Sin (II), Tan (III), Cos (IV) positive — reciprocals share signs.
  • Domain-range: sin, cos: R→[−1,1]\mathbb{R} \to [-1, 1]. tan, cot: all reals as outputs; tan undefined at odd multiples of π2\frac{\pi}{2}, cot at multiples of π\pi. sec, cosec: outputs outside (−1,1)(-1, 1).

The Identity Machine — Formula Card

Sum-difference:

  • cos⁡(x±y)=cos⁡xcos⁡y∓sin⁡xsin⁡y\cos(x \pm y) = \cos x\cos y \mp \sin x\sin y (sign flips)
  • sin⁡(x±y)=sin⁡xcos⁡y±cos⁡xsin⁡y\sin(x \pm y) = \sin x\cos y \pm \cos x\sin y (sign keeps)
  • tan⁡(x±y)=tan⁡x±tan⁡y1∓tan⁡xtan⁡y\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x\tan y}; cot⁡(x+y)=cot⁡xcot⁡y−1cot⁡y+cot⁡x\cot(x + y) = \frac{\cot x\cot y - 1}{\cot y + \cot x}

Allied angles: with π\pi, 2π2\pi the function survives; with π2\frac{\pi}{2}, 3π2\frac{3\pi}{2} it swaps; sign from the original angle's quadrant. Key rows: sin⁡(π−x)=sin⁡x\sin(\pi - x) = \sin x, cos⁡(π+x)=−cos⁡x\cos(\pi + x) = -\cos x, sin⁡(π2+x)=cos⁡x\sin\left(\frac{\pi}{2} + x\right) = \cos x.

Double angle: sin⁡2x=2sin⁡xcos⁡x=2tan⁡x1+tan⁡2x\sin 2x = 2\sin x\cos x = \frac{2\tan x}{1 + \tan^2 x}; cos⁡2x=cos⁡2x−sin⁡2x=2cos⁡2x−1=1−2sin⁡2x=1−tan⁡2x1+tan⁡2x\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x = \frac{1 - \tan^2 x}{1 + \tan^2 x}; tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2\tan x}{1 - \tan^2 x}.

Power reduction: cos⁡2x=1+cos⁡2x2\cos^2 x = \frac{1 + \cos 2x}{2}, sin⁡2x=1−cos⁡2x2\sin^2 x = \frac{1 - \cos 2x}{2}; and 1±sin⁡2x=(sin⁡x±cos⁡x)21 \pm \sin 2x = (\sin x \pm \cos x)^2.

Triple angle: sin⁡3x=3sin⁡x−4sin⁡3x\sin 3x = 3\sin x - 4\sin^3 x; cos⁡3x=4cos⁡3x−3cos⁡x\cos 3x = 4\cos^3 x - 3\cos x; tan⁡3x=3tan⁡x−tan⁡3x1−3tan⁡2x\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}.

Useful squares: sin⁡2A−sin⁡2B=sin⁡(A+B)sin⁡(A−B)\sin^2 A - \sin^2 B = \sin(A+B)\sin(A-B); cos⁡(x+y)cos⁡(x−y)=cos⁡2x−sin⁡2y\cos(x+y)\cos(x-y) = \cos^2 x - \sin^2 y.

Transformations and Special Values — Formula Card

Sums → products:

  • cos⁡x+cos⁡y=2cos⁡x+y2cos⁡x−y2\cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2}; cos⁡x−cos⁡y=−2sin⁡x+y2sin⁡x−y2\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2} (the minus!)
  • sin⁡x+sin⁡y=2sin⁡x+y2cos⁡x−y2\sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}; sin⁡x−sin⁡y=2cos⁡x+y2sin⁡x−y2\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}

Products → sums: 2cos⁡xcos⁡y=cos⁡(x+y)+cos⁡(x−y)2\cos x\cos y = \cos(x+y) + \cos(x-y); −2sin⁡xsin⁡y=cos⁡(x+y)−cos⁡(x−y)-2\sin x\sin y = \cos(x+y) - \cos(x-y); 2sin⁡xcos⁡y=sin⁡(x+y)+sin⁡(x−y)2\sin x\cos y = \sin(x+y) + \sin(x-y).

Special values: sin⁡15°=cos⁡75°=3−122\sin 15° = \cos 75° = \frac{\sqrt{3}-1}{2\sqrt{2}}; tan⁡15°=2−3\tan 15° = 2 - \sqrt{3}; tan⁡75°=2+3\tan 75° = 2 + \sqrt{3}; tan⁡22.5°=2−1\tan 22.5° = \sqrt{2} - 1; sin⁡18°=5−14\sin 18° = \frac{\sqrt{5}-1}{4}; cos⁡36°=5+14\cos 36° = \frac{\sqrt{5}+1}{4}; sin⁡18°cos⁡36°=14\sin 18°\cos 36° = \frac{1}{4}.

Standard products: cos⁡20°cos⁡40°cos⁡80°=18\cos 20°\cos 40°\cos 80° = \frac{1}{8}; sin⁡θsin⁡(60°−θ)sin⁡(60°+θ)=sin⁡3θ4\sin\theta\sin(60°-\theta)\sin(60°+\theta) = \frac{\sin 3\theta}{4}; ∏k=0n−1cos⁡2kx=sin⁡2nx2nsin⁡x\prod_{k=0}^{n-1}\cos 2^k x = \frac{\sin 2^n x}{2^n\sin x}.

JEE Equations and Ranges — Quick Card

General solutions (n∈Zn \in \mathbb{Z}):

  • sin⁡x=sin⁡y\sin x = \sin y: x=nπ+(−1)nyx = n\pi + (-1)^n y; cos⁡x=cos⁡y\cos x = \cos y: x=2nπ±yx = 2n\pi \pm y; tan⁡x=tan⁡y\tan x = \tan y: x=nπ+yx = n\pi + y
  • sin x = 0: nπn\pi; cos x = 0: (2n+1)π2(2n+1)\frac{\pi}{2}; sin x = 1: 2nπ+π22n\pi + \frac{\pi}{2}; cos x = 1: 2nπ2n\pi
  • Protocol: reduce to (function) = (value) by FACTORING (never divide by a vanishing factor), then quote the master formula; check with n = 0, 1.

The asin⁡x+bcos⁡xa\sin x + b\cos x machine: =Rsin⁡(x+ϕ)= R\sin(x + \phi), R=a2+b2R = \sqrt{a^2 + b^2}; range [−R,R][-R, R]; asin⁡x+bcos⁡x=ca\sin x + b\cos x = c solvable iff ∣c∣≤R|c| \leq R; with +c shift, range [c−R,c+R][c - R, c + R].

Quadratic-in-sine ranges: substitute t = sin x ∈[−1,1]\in [-1, 1], analyse the parabola on the interval (vertex if inside + both endpoints).

Conditional identities (A + B + C = π\pi): sin(A+B) = sin C, cos(A+B) = −cos⁡C-\cos C, sin⁡A+B2=cos⁡C2\sin\frac{A+B}{2} = \cos\frac{C}{2}. Catalogue: tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A + \tan B + \tan C = \tan A\tan B\tan C; sin⁡2A+sin⁡2B+sin⁡2C=4sin⁡Asin⁡Bsin⁡C\sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C; cos⁡A+cos⁡B+cos⁡C=1+4sin⁡A2sin⁡B2sin⁡C2\cos A + \cos B + \cos C = 1 + 4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}; pairwise half-angle tangent products sum to 1.

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by thousands of students every year:

  1. l=rθl = r\theta needs radians — convert degrees first, every time.
  2. A bare number is radians: sin 2 means 2 radians (≈\approx 114.6°), not 2°.
  3. Sign before value: announce the quadrant, apply ASTC, THEN write the magnitude.
  4. [x][x]-style slips here: sin⁡(π−x)=+sin⁡x\sin(\pi - x) = +\sin x but cos⁡(π−x)=−cos⁡x\cos(\pi - x) = -\cos x — the pair is asymmetric.
  5. tan and cot have period π\pi, not 2π2\pi — their general solutions carry nπn\pi.
  6. cos⁡x−cos⁡y\cos x - \cos y starts with −2sin⁡-2\sin — the only transformation formula with a leading minus.
  7. Max of asin⁡x+bcos⁡xa\sin x + b\cos x is a2+b2\sqrt{a^2 + b^2}, never a + b — the two extremes cannot coincide.
  8. Never divide an equation by sin x or cos x — factor instead, or a whole solution family dies.
  9. Reject impossible roots (sin⁡x=2\sin x = 2) with the reason ∣sin⁡x∣≤1|\sin x| \leq 1 — the rejection carries a mark.
  10. Half-angle signs: locate the quadrant of x2\frac{x}{2} from the interval of x BEFORE taking square roots in power reduction.

Done revising? Take the Section 8 mock drill under exam timing — that is the real test of readiness.