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MathematicsTrigonometryHigh Yield (2-3 Qs in JEE)

Trigonometric Ratios & Identities Formulas

Compound angles, transformation formulas, multiple angles, conditional identities, and general trigonometric equations.

10 Formulas•426+ Questions in Bank

Concept Diagram

Trigonometric Unit Circle
XYQ-I: All (+)Q-II: Sin (+)Q-III: Tan (+)Q-IV: Cos (+)P(cos θ, sin θ)θ(1, 0)(0, 1)
Standard angles (0, π/6, π/4, π/3, π/2), (cos θ, sin θ) coordinates, and ASTC quadrant sign rules.

Core Formulas & Mathematical Identities

10
01

Compound Angle Formulas

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B,cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B, \quad \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
Diagram Reference:Compound Angle Formulas

Fundamental addition and subtraction identities.

Examiner Tip: Notice the minus-plus inversion in cos(A ± B).
02

Transformation (Product to Sum & Sum to Product)

2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B),cos⁡C+cos⁡D=2cos⁡C+D2cos⁡C−D22\sin A\cos B = \sin(A+B) + \sin(A-B), \quad \cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}

Converts trigonometric products into sums and vice versa for easy calculus integration.

Examiner Tip: cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2) = 2 sin((C+D)/2) sin((D-C)/2).
03

Multiple Angle Formulas (Double & Half)

sin⁡2A=2tan⁡A1+tan⁡2A,cos⁡2A=1−tan⁡2A1+tan⁡2A=2cos⁡2A−1=1−2sin⁡2A\sin 2A = \frac{2\tan A}{1 + \tan^2 A}, \quad \cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A} = 2\cos^2 A - 1 = 1 - 2\sin^2 A

Expresses double angle functions entirely in terms of tan A.

Examiner Tip: 1 - cos 2A = 2 sin²A, and 1 + cos 2A = 2 cos²A.
04

Maximum and Minimum of a sin θ + b cos θ

−a2+b2≤asin⁡θ+bcos⁡θ≤a2+b2-\sqrt{a^2 + b^2} \le a\sin\theta + b\cos\theta \le \sqrt{a^2 + b^2}

Extremum values of linear harmonic combination of sine and cosine.

Examiner Tip: Useful for range questions in both Mathematics and Physics SHM.
05

General Solutions of Trigonometric Equations

sin⁡θ=sin⁡α  ⟹  θ=nπ+(−1)nα,cos⁡θ=cos⁡α  ⟹  θ=2nπ±α,tan⁡θ=tan⁡α  ⟹  θ=nπ+α\sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n \alpha, \quad \cos\theta = \cos\alpha \implies \theta = 2n\pi \pm \alpha, \quad \tan\theta = \tan\alpha \implies \theta = n\pi + \alpha

Fundamental general solutions for periodic trigonometric equations (where n ∈ ℤ).

Examiner Tip: For a·cos θ + b·sin θ = c to have real solutions, condition is |c| ≤ √(a² + b²).
06

Quadratic & Squared Trigonometric Equations

sin⁡2θ=sin⁡2α  ⟺  cos⁡2θ=cos⁡2α  ⟺  tan⁡2θ=tan⁡2α  ⟹  θ=nπ±α\sin^2\theta = \sin^2\alpha \iff \cos^2\theta = \cos^2\alpha \iff \tan^2\theta = \tan^2\alpha \implies \theta = n\pi \pm \alpha

Universal general solution for all squared trigonometric equation forms.

Examiner Tip: Always check for extraneous roots when squaring both sides of a trigonometric equation.
07

Triple Angle Identities

sin⁡3A=3sin⁡A−4sin⁡3A,cos⁡3A=4cos⁡3A−3cos⁡A,tan⁡3A=3tan⁡A−tan⁡3A1−3tan⁡2A\sin 3A = 3\sin A - 4\sin^3 A, \quad \cos 3A = 4\cos^3 A - 3\cos A, \quad \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}

Expansion of triple angles, frequently applied in cubic root and polynomial evaluations.

Examiner Tip: tan A · tan(60° - A) · tan(60° + A) = tan 3A, and sin A · sin(60° - A) · sin(60° + A) = (1/4) sin 3A.
08

Summation of Sines & Cosines in A.P.

∑k=0n−1sin⁡(α+kβ)=sin⁡(nβ/2)sin⁡(β/2)sin⁡(α+(n−1)β2),∑k=0n−1cos⁡(α+kβ)=sin⁡(nβ/2)sin⁡(β/2)cos⁡(α+(n−1)β2)\sum_{k=0}^{n-1} \sin(\alpha + k\beta) = \frac{\sin(n\beta/2)}{\sin(\beta/2)} \sin\left(\alpha + \frac{(n-1)\beta}{2}\right), \quad \sum_{k=0}^{n-1} \cos(\alpha + k\beta) = \frac{\sin(n\beta/2)}{\sin(\beta/2)} \cos\left(\alpha + \frac{(n-1)\beta}{2}\right)

Closed-form summation for harmonic series whose angles progress in an arithmetic sequence.

Examiner Tip: Crucial for interference of n waves in Physics Wave Optics as well as JEE Advanced Maths.
09

Conditional Identities in Triangles (A + B + C = π)

tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C,sin⁡2A+sin⁡2B+sin⁡2C=4sin⁡Asin⁡Bsin⁡C,cos⁡2A+cos⁡2B+cos⁡2C=1−2cos⁡Acos⁡Bcos⁡C\tan A + \tan B + \tan C = \tan A \tan B \tan C, \quad \sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C, \quad \cos^2 A + \cos^2 B + \cos^2 C = 1 - 2\cos A\cos B\cos C

Standard cyclic identities holding inside any Euclidean triangle.

Examiner Tip: If A + B + C = π/2, then tan A tan B + tan B tan C + tan C tan A = 1.
10

Heights & Distances Shortcut Formulas

h=dcot⁡α−cot⁡β=dsin⁡αsin⁡βsin⁡(β−α),Hcloud=h⋅tan⁡β+tan⁡αtan⁡β−tan⁡αh = \frac{d}{\cot\alpha - \cot\beta} = \frac{d\sin\alpha\sin\beta}{\sin(\beta - \alpha)}, \quad H_{\text{cloud}} = h\cdot\frac{\tan\beta + \tan\alpha}{\tan\beta - \tan\alpha}

Height of tower h from two collinear points distance d apart with elevation angles α and β (β > α). Cloud height H above lake from observer at height h above water with elevation α and reflection depression β.

Examiner Tip: Tower between two observation points: h = d / (cot α + cot β). Flagstaff of height p on top of tower: tower height h = p tan α / (tan β - tan α).

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Trigonometric Ratios & Identities

  • •Forgetting the ± sign or quadrant ASTC rules when taking square roots of 1 - cos²θ.
  • •General solution for sin θ = 0 is θ = nπ, while for cos θ = 0 it is θ = (2n + 1)π/2.
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