Math · Trigonometry

Trigonometric Functions formulas for JEE

Every Trigonometric Functions formula you need for JEE, grouped by concept.

FormulasRevision notes
56 formulas4 concepts
01Trigonometric Functions & Angles02Trigonometric Identities03Trigonometric Equations04Properties of Triangles
01

Trigonometric Functions & Angles

15 formulas

Arc Length

l=rθl = r\theta

Length of an arc subtending a central angle.

applies whenθ\theta must be in radians.
arc lengthradians

Cosine Cofunction Identity

cos⁡(π2−x)=sin⁡x\cos(\frac{\pi}{2} - x) = \sin x

Cosine of complementary angle.

cofunctioncomplementary

Sine Cofunction Identity

sin⁡(π2−x)=cos⁡x\sin(\frac{\pi}{2} - x) = \cos x

Sine of complementary angle.

cofunctioncomplementary

Cosine of Negative Angle

cos⁡(−x)=cos⁡x\cos(-x) = \cos x

Even function property of cosine.

even functionnegative angle

Cotangent-Cosecant Identity

1+cot⁡2x=cosec2x1 + \cot^2 x = \text{cosec}^2 x

Identity linking cotangent and cosecant.

applies whenx≠nπx \neq n\pi
fundamentalidentity

Cotangent Definition

cot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}

Cotangent in terms of cosine and sine.

applies whenx≠nπx \neq n\pi
definitionfundamental

Cosecant Definition

cosec x=1sin⁡x\text{cosec } x = \frac{1}{\sin x}

Cosecant as reciprocal of sine.

applies whenx≠nπx \neq n\pi
definitionfundamental

Pythagorean Identity

cos⁡2x+sin⁡2x=1\cos^2 x + \sin^2 x = 1

Fundamental trigonometric identity.

fundamentalidentity

Radian to Degree Conversion

Radian measure=π180×Degree measure\text{Radian measure} = \frac{\pi}{180} \times \text{Degree measure}

Relation between radian and degree measures.

conversionunits

Secant Definition

sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}

Secant as reciprocal of cosine.

applies whenx≠(2n+1)π2x \neq (2n+1)\frac{\pi}{2}
definitionfundamental

Sine of Negative Angle

sin⁡(−x)=−sin⁡x\sin(-x) = -\sin x

Odd function property of sine.

odd functionnegative angle

Cosine of Supplementary Angle

cos⁡(π−x)=−cos⁡x\cos(\pi - x) = -\cos x

Cosine of pi minus x.

supplementary

Sine of Supplementary Angle

sin⁡(π−x)=sin⁡x\sin(\pi - x) = \sin x

Sine of pi minus x.

supplementary

Tangent Definition

tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}

Tangent in terms of sine and cosine.

applies whenx≠(2n+1)π2x \neq (2n+1)\frac{\pi}{2}
definitionfundamental

Tangent-Secant Identity

1+tan⁡2x=sec⁡2x1 + \tan^2 x = \sec^2 x

Identity linking tangent and secant.

applies whenx≠(2n+1)π2x \neq (2n+1)\frac{\pi}{2}
fundamentalidentity
02

Trigonometric Identities

32 formulas

Conditional Identity (Sine)

sin⁡2A+sin⁡2B+sin⁡2C=4sin⁡Asin⁡Bsin⁡C\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C

Sum of sines in a triangle.

applies whenA+B+C=πA + B + C = \pi
jee-advancedconditional identity

Conditional Identity (Tangent)

tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A + \tan B + \tan C = \tan A \tan B \tan C

Sum of tangents in a triangle.

applies whenA+B+C=πA + B + C = \pi
jee-advancedconditional identity

Cosine Angle Difference

cos⁡(x−y)=cos⁡xcos⁡y+sin⁡xsin⁡y\cos(x - y) = \cos x \cos y + \sin x \sin y

Cosine of the difference of two angles.

compound angledifference

Cosine Double Angle

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}

Double angle identity for cosine.

applies whenFor tan variant, x≠(2n+1)π2x \neq (2n+1)\frac{\pi}{2}
double angle

Cosine Angle Sum

cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos(x + y) = \cos x \cos y - \sin x \sin y

Cosine of the sum of two angles.

compound anglesum

Cosine Triple Angle

cos⁡3x=4cos⁡3x−3cos⁡x\cos 3x = 4\cos^3 x - 3\cos x

Triple angle identity for cosine.

triple angle

Cotangent Angle Difference

cot⁡(x−y)=cot⁡xcot⁡y+1cot⁡y−cot⁡x\cot(x - y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}

Cotangent of the difference of two angles.

applies whenNone of x,y,x−yx, y, x-y is a multiple of π\pi
compound angledifference

Cotangent Angle Sum

cot⁡(x+y)=cot⁡xcot⁡y−1cot⁡y+cot⁡x\cot(x + y) = \frac{\cot x \cot y - 1}{\cot y + \cot x}

Cotangent of the sum of two angles.

applies whenNone of x,y,x+yx, y, x+y is a multiple of π\pi
compound anglesum

Extrema of Linear Combination

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

Maximum and minimum values of a sin x + b cos x.

jee-advancedrange

Cosine Quadruple Angle

cos⁡4x=1−8sin⁡2xcos⁡2x\cos 4x = 1 - 8\sin^2 x \cos^2 x

Expansion of cosine 4x.

multiple angleexercise

Cosine Sextuple Angle

cos⁡6x=32cos⁡6x−48cos⁡4x+18cos⁡2x−1\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1

Expansion of cosine 6x.

multiple angleexercise

Cosine Product Series

∏k=0n−1cos⁡(2kx)=sin⁡(2nx)2nsin⁡x\prod_{k=0}^{n-1} \cos(2^k x) = \frac{\sin(2^n x)}{2^n \sin x}

Product of cosines with doubling angles.

applies whenx≠mπx \neq m\pi
jee-advancedproduct series

Product to Sum (Cos Cos)

2cos⁡xcos⁡y=cos⁡(x+y)+cos⁡(x−y)2\cos x \cos y = \cos(x+y) + \cos(x-y)

Transformation of product of cosines into sum.

product to sumtransformation

Product to Sum (Cos Sin)

2cos⁡xsin⁡y=sin⁡(x+y)−sin⁡(x−y)2\cos x \sin y = \sin(x+y) - \sin(x-y)

Transformation of cosine and sine product into difference.

product to sumtransformation

Product to Sum (Sin Cos)

2sin⁡xcos⁡y=sin⁡(x+y)+sin⁡(x−y)2\sin x \cos y = \sin(x+y) + \sin(x-y)

Transformation of sine and cosine product into sum.

product to sumtransformation

Product to Sum (Sin Sin)

−2sin⁡xsin⁡y=cos⁡(x+y)−cos⁡(x−y)-2\sin x \sin y = \cos(x+y) - \cos(x-y)

Transformation of product of sines into sum/difference.

product to sumtransformation

Cosine Series in A.P.

∑k=0n−1cos⁡(α+kβ)=sin⁡(nβ2)sin⁡(β2)cos⁡(α+(n−1)β2)\sum_{k=0}^{n-1} \cos(\alpha + k\beta) = \frac{\sin(\frac{n\beta}{2})}{\sin(\frac{\beta}{2})} \cos(\alpha + (n-1)\frac{\beta}{2})

Sum of cosines of angles in arithmetic progression.

applies whenβ≠2mπ\beta \neq 2m\pi
jee-advancedseries

Sine Series in A.P.

∑k=0n−1sin⁡(α+kβ)=sin⁡(nβ2)sin⁡(β2)sin⁡(α+(n−1)β2)\sum_{k=0}^{n-1} \sin(\alpha + k\beta) = \frac{\sin(\frac{n\beta}{2})}{\sin(\frac{\beta}{2})} \sin(\alpha + (n-1)\frac{\beta}{2})

Sum of sines of angles in arithmetic progression.

applies whenβ≠2mπ\beta \neq 2m\pi
jee-advancedseries

Sine Angle Difference

sin⁡(x−y)=sin⁡xcos⁡y−cos⁡xsin⁡y\sin(x - y) = \sin x \cos y - \cos x \sin y

Sine of the difference of two angles.

compound angledifference

Sine 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}

Double angle identity for sine.

applies whenFor tan variant, x≠(2n+1)π2x \neq (2n+1)\frac{\pi}{2}
double angle

Sine Angle Sum

sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin(x + y) = \sin x \cos y + \cos x \sin y

Sine of the sum of two angles.

compound anglesum

Sine Triple Angle

sin⁡3x=3sin⁡x−4sin⁡3x\sin 3x = 3\sin x - 4\sin^3 x

Triple angle identity for sine.

triple angle

Sum to Product (Cosine Add)

cos⁡x+cos⁡y=2cos⁡x+y2cos⁡x−y2\cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2}

Transformation of cosine sum into product.

sum to producttransformation

Sum to Product (Cosine Subtract)

cos⁡x−cos⁡y=−2sin⁡x+y2sin⁡x−y2\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}

Transformation of cosine difference into product.

sum to producttransformation

Sum to Product (Sine Add)

sin⁡x+sin⁡y=2sin⁡x+y2cos⁡x−y2\sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}

Transformation of sine sum into product.

sum to producttransformation

Sum to Product (Sine Subtract)

sin⁡x−sin⁡y=2cos⁡x+y2sin⁡x−y2\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}

Transformation of sine difference into product.

sum to producttransformation

Tangent Angle Difference

tan⁡(x−y)=tan⁡x−tan⁡y1+tan⁡xtan⁡y\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}

Tangent of the difference of two angles.

applies whenNone of x,y,x−yx, y, x-y is an odd multiple of π2\frac{\pi}{2}
compound angledifference

Tangent Double Angle

tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2\tan x}{1 - \tan^2 x}

Double angle identity for tangent.

applies whenx≠(2n+1)π4x \neq (2n+1)\frac{\pi}{4} and x≠(2n+1)π2x \neq (2n+1)\frac{\pi}{2}
double angle

Tangent Angle Sum

tan⁡(x+y)=tan⁡x+tan⁡y1−tan⁡xtan⁡y\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}

Tangent of the sum of two angles.

applies whenNone of x,y,x+yx, y, x+y is an odd multiple of π2\frac{\pi}{2}
compound anglesum

Tangent Triple Angle

tan⁡3x=3tan⁡x−tan⁡3x1−3tan⁡2x\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}

Triple angle identity for tangent.

applies when3x≠(2n+1)π23x \neq (2n+1)\frac{\pi}{2}
triple angle

Sine of 15 degrees

sin⁡15∘=3−122\sin 15^\circ = \frac{\sqrt{3}-1}{2\sqrt{2}}

Exact value of sine 15 degrees.

exact valueworked example

Tangent of 22.5 degrees

tan⁡π8=2−1\tan \frac{\pi}{8} = \sqrt{2} - 1

Exact value of tangent pi/8.

exact valueworked example
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03

Trigonometric Equations

5 formulas

General Solution Cosine

cos⁡x=cos⁡y  ⟹  x=2nπ±y\cos x = \cos y \implies x = 2n\pi \pm y

General solution for cosine equations.

applies whenn∈Zn \in \mathbb{Z}
jee-advancedequation

General Solution Sine

sin⁡x=sin⁡y  ⟹  x=nπ+(−1)ny\sin x = \sin y \implies x = n\pi + (-1)^n y

General solution for sine equations.

applies whenn∈Zn \in \mathbb{Z}
jee-advancedequation

General Solution Tangent

tan⁡x=tan⁡y  ⟹  x=nπ+y\tan x = \tan y \implies x = n\pi + y

General solution for tangent equations.

applies whenn∈Zn \in \mathbb{Z}
jee-advancedequation

Zero Solution Cosine

cos⁡x=0  ⟹  x=(2n+1)π2\cos x = 0 \implies x = (2n+1)\frac{\pi}{2}

Roots of the cosine function.

applies whenn∈Zn \in \mathbb{Z}
equationroots

Zero Solution Sine

sin⁡x=0  ⟹  x=nπ\sin x = 0 \implies x = n\pi

Roots of the sine function.

applies whenn∈Zn \in \mathbb{Z}
equationroots
04

Properties of Triangles

4 formulas

Cosine Rule

cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}

Relating sides and one angle of a triangle.

jee-advancedproperties of triangles

Napier's Analogy

tan⁡A−B2=a−ba+bcot⁡C2\tan\frac{A-B}{2} = \frac{a-b}{a+b}\cot\frac{C}{2}

Tangent of half angle difference.

jee-advancedproperties of triangles

Projection Rule

a=bcos⁡C+ccos⁡Ba = b\cos C + c\cos B

Side length in terms of other sides and angles.

jee-advancedproperties of triangles

Sine Rule

asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Proportionality of sides and sines of opposite angles.

applies whenRR is the circumradius
jee-advancedproperties of triangles
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