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Trigonometry Formulas

Complete chapter-wise formula sheet for Trigonometry (Mathematics) covering key equations, parameters, traps, and concept diagrams for JEE Main & Advanced.

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Chapter Directory (26 Chapters)

Click any chapter to view dedicated formula sheet & concept diagrams
Trigonometry426 Qs

Trigonometric Ratios & Identities

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

10 Formulas Diagram
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Trigonometry470 Qs

Inverse Trigonometric Functions (ITF)

Principal branches, domain & range, complementary angle relations, and sum of inverse tangents.

8 Formulas Diagram
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Trigonometry351 Qs

Properties of Triangles (SOT)

Sine rule, cosine rule, projection formula, radii of circumcircle, incircle, and excircles.

8 Formulas
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Formulas Reference (Trigonometry)

3 Chapters match
TrigonometryHigh Yield (2-3 Qs in JEE)426 PYQs

Trigonometric Ratios & Identities

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

Chapter

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

Fundamental addition and subtraction identities.

Notice the minus-plus inversion in cos(A ± B).

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.

cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2) = 2 sin((C+D)/2) sin((D-C)/2).

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.

1 - cos 2A = 2 sin²A, and 1 + cos 2A = 2 cos²A.

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.

Useful for range questions in both Mathematics and Physics SHM.

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).

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

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.

Always check for extraneous roots when squaring both sides of a trigonometric equation.

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.

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

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.

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

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.

If A + B + C = π/2, then tan A tan B + tan B tan C + tan C tan A = 1.

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.

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).
TrigonometryHigh Yield (1-2 Qs in JEE)470 PYQs

Inverse Trigonometric Functions (ITF)

Principal branches, domain & range, complementary angle relations, and sum of inverse tangents.

Chapter

Principal Value Domains & Ranges

sin⁡−1x∈[−π2,π2],cos⁡−1x∈[0,π],tan⁡−1x∈(−π2,π2)\sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right], \quad \cos^{-1}x \in [0, \pi], \quad \tan^{-1}x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

Standard principal branches defined to make trigonometric functions bijective.

Always reduce angles outside principal branch into the fundamental domain first.

Complementary Angles Property

sin⁡−1x+cos⁡−1x=π2,tan⁡−1x+cot⁡−1x=π2(x∈Domain)\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \quad (x \in \text{Domain})

Sum of co-functions equals π/2 everywhere in their common domain.

sec⁻¹x + csc⁻¹x = π/2 for |x| ≥ 1.

Sum of Inverse Tangents

tan⁡−1x+tan⁡−1y={tan⁡−1(x+y1−xy)xy<1π+tan⁡−1(x+y1−xy)x,y>0,  xy>1\tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\left(\frac{x+y}{1-xy}\right) & xy < 1 \\ \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right) & x,y > 0, \; xy > 1 \end{cases}

Combines two inverse tangents into one.

Verify the sign of xy before applying the formula without π adjustment.

Multiple Angle Inverse Formula

2tan⁡−1x=sin⁡−1(2x1+x2)=cos⁡−1(1−x21+x2)=tan⁡−1(2x1−x2)2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right)

Interconversion between 2 tan⁻¹x and other inverse functions.

Check domain conditions: cos⁻¹((1-x²)/(1+x²)) = 2 tan⁻¹x holds for x ≥ 0.

Negative Argument Relations

sin⁡−1(−x)=−sin⁡−1x,cos⁡−1(−x)=π−cos⁡−1x,tan⁡−1(−x)=−tan⁡−1x\sin^{-1}(-x) = -\sin^{-1}x, \quad \cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x

Odd vs supplementary behavior for negative arguments.

sec⁻¹(-x) = π - sec⁻¹x and cot⁻¹(-x) = π - cot⁻¹x.

Telescoping Series of Inverse Tangents

∑r=1ntan⁡−1(xr−xr−11+xrxr−1)=tan⁡−1xn−tan⁡−1x0,tan⁡−1(11+r+r2)=tan⁡−1(r+1)−tan⁡−1r\sum_{r=1}^n \tan^{-1}\left(\frac{x_r - x_{r-1}}{1 + x_r x_{r-1}}\right) = \tan^{-1}x_n - \tan^{-1}x_0, \quad \tan^{-1}\left(\frac{1}{1 + r + r^2}\right) = \tan^{-1}(r+1) - \tan^{-1}r

Decomposition of rational algebraic arguments into differences of consecutive inverse tangents for infinite summation.

For ∑_{r=1}^∞ tan⁻¹(1/(r² + r + 1)): Sum = lim_{n→∞} (tan⁻¹(n+1) - tan⁻¹ 1) = π/2 - π/4 = π/4.

Addition Formulas for Inverse Sine & Cosine

sin⁡−1x±sin⁡−1y=sin⁡−1(x1−y2±y1−x2),cos⁡−1x±cos⁡−1y=cos⁡−1(xy∓1−x21−y2)\sin^{-1}x \pm \sin^{-1}y = \sin^{-1}\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right), \quad \cos^{-1}x \pm \cos^{-1}y = \cos^{-1}\left(xy \mp \sqrt{1-x^2}\sqrt{1-y^2}\right)

Standard addition and subtraction formulas for inverse sines and cosines within primary domains.

For sin⁻¹x + sin⁻¹y: if x² + y² > 1 and x,y > 0, the sum exceeds π/2 and equals π - sin⁻¹(x√(1-y²) + y√(1-x²)).

Extreme Values of Inverse Trigonometric Sums

(sin⁡−1x)2+(cos⁡−1x)2∈[π28,5π24],(sin⁡−1x)3+(cos⁡−1x)3∈[π332,7π38](\sin^{-1}x)^2 + (\cos^{-1}x)^2 \in \left[\frac{\pi^2}{8}, \frac{5\pi^2}{4}\right], \quad (\sin^{-1}x)^3 + (\cos^{-1}x)^3 \in \left[\frac{\pi^3}{32}, \frac{7\pi^3}{8}\right]

Ranges of quadratic and cubic symmetric combinations of inverse sine and cosine.

Minimum occurs when sin⁻¹x = cos⁻¹x = π/4 (i.e. x = 1/√2). Maximum occurs at x = -1.
TrigonometryHigh Yield (1 Q in JEE)351 PYQs

Properties of Triangles (SOT)

Sine rule, cosine rule, projection formula, radii of circumcircle, incircle, and excircles.

Chapter

Sine Rule & Circumradius R

asin⁡A=bsin⁡B=csin⁡C=2R,R=abc4Δ\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, \quad R = \frac{abc}{4\Delta}

Connects sides, angles, and circumradius R of triangle ABC.

Δ is the area of triangle: Δ = (1/2)ab sin C.

Cosine Rule

cos⁡A=b2+c2−a22bc,cos⁡B=c2+a2−b22ca,cos⁡C=a2+b2−c22ab\cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}

Determines angles when all three sides are known.

If a² = b² + c², then A = 90° (Pythagoras theorem).

Inradius & Exradii Formulas

r=Δs=4Rsin⁡A2sin⁡B2sin⁡C2,r1=Δs−a=4Rsin⁡A2cos⁡B2cos⁡C2r = \frac{\Delta}{s} = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \quad r_1 = \frac{\Delta}{s - a} = 4R\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}

Radii of incircle and excircles (opposite to vertices A, B, C).

1/r₁ + 1/r₂ + 1/r₃ = 1/r and r₁ r₂ r₃ = s Δ.

Napier's Analogy (Law of Tangents)

tan⁡(B−C2)=b−cb+ccot⁡A2,tan⁡(C−A2)=c−ac+acot⁡B2,tan⁡(A−B2)=a−ba+bcot⁡C2\tan\left(\frac{B - C}{2}\right) = \frac{b - c}{b + c}\cot\frac{A}{2}, \quad \tan\left(\frac{C - A}{2}\right) = \frac{c - a}{c + a}\cot\frac{B}{2}, \quad \tan\left(\frac{A - B}{2}\right) = \frac{a - b}{a + b}\cot\frac{C}{2}

Connects difference of two angles to the ratio of side lengths and half the third angle.

Often utilized in JEE Advanced when side differences (b - c) or angle differences are given.

Half-Angle Formulas (Sides & Semi-Perimeter)

sin⁡A2=(s−b)(s−c)bc,cos⁡A2=s(s−a)bc,tan⁡A2=(s−b)(s−c)s(s−a)=Δs(s−a)\sin\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}, \quad \cos\frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}}, \quad \tan\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} = \frac{\Delta}{s(s-a)}

Trigonometric half-angle ratios expressed purely in terms of side lengths and semi-perimeter s.

Area of triangle via Heron's formula is Δ = √(s(s-a)(s-b)(s-c)).

Distances Between Triangle Centers & Euler Line

OI2=R2−2Rr  (Euler’s Incenter Formula),HG:GO=2:1  (Euler Line)OI^2 = R^2 - 2Rr \; (\text{Euler's Incenter Formula}), \quad HG : GO = 2 : 1 \; (\text{Euler Line})

Metric distance between circumcenter O and incenter I, and collinearity ratio of Orthocenter (H), Centroid (G), and Circumcenter (O).

In an equilateral triangle, H, G, I, and O all coincide into a single point, giving R = 2r.

Projection Formulas & Median Length (Apollonius)

a=bcos⁡C+ccos⁡B,ma=122b2+2c2−a2,ma2+mb2+mc2=34(a2+b2+c2)a = b\cos C + c\cos B, \quad m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}, \quad m_a^2 + m_b^2 + m_c^2 = \frac{3}{4}(a^2 + b^2 + c^2)

Projection of sides onto each other, and closed-form length of median from vertex A bisecting opposite side a.

Centroid divides each median in 2:1 ratio from vertex: AG = (2/3) m_a.

m-n Cotangent Theorem

(m+n)cot⁡θ=mcot⁡α−ncot⁡β=ncot⁡B−mcot⁡C(m + n)\cot\theta = m\cot\alpha - n\cot\beta = n\cot B - m\cot C

Relates angle θ formed by a cevian dividing the base into segments m : n to the base angles or divided vertex angles α and β.

A staple shortcut in JEE Advanced geometry questions involving internal transversals.