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Trigonometry Formula Sheet for JEE

80+ JEE formulas in this unit

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The Trigonometry JEE formula sheet lists 80+ important formulas for JEE Main and Advanced, including essential identities from Trigonometrical identities and trigonometrical functions, Inverse trigonometrical functions and their properties. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Trigonometry. This unit-wise formula list covers 80+ exam-relevant results across Trigonometrical identities and trigonometrical functions, Inverse trigonometrical functions and their properties, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

80 formulas across 2 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit
Essential: 44Important: 30Supplementary: 6

Trigonometrical identities and trigonometrical functions

Radian measure of an angleImportant
θ (in radians)=lr\theta\text{ (in radians)}=\dfrac{l}{r}
Variables
ll = length of subtended arc, rr = radius.
Conditions
For a circle of radius rr.
Where used in JEE
Conversion between arc length, sector area, and trigonometric arguments in radians.
Degree-radian conversionEssential
π rad=180∘,1∘=π180 rad,1 rad=180∘π\pi\text{ rad}=180^\circ,\quad 1^\circ=\dfrac{\pi}{180}\text{ rad},\quad 1\text{ rad}=\dfrac{180^\circ}{\pi}
Where used in JEE
Angle conversion; evaluating trigonometric functions and limits.
Arc length formulaImportant
l=rθl=r\theta
Variables
ll = arc length, rr = radius, θ\theta = angle in radians.
Conditions
θ\theta must be in radians.
Where used in JEE
Geometry-based trigonometry applications.
Area of sectorImportant
A=12r2θA=\dfrac{1}{2}r^2\theta
Variables
AA = area of sector, rr = radius, θ\theta = angle in radians.
Conditions
θ\theta must be in radians.
Where used in JEE
Circular geometry and trigonometric applications.
Trigonometric ratios in a right triangleEssential
sin⁡θ=perpendicularhypotenuse,cos⁡θ=basehypotenuse,tan⁡θ=perpendicularbase\sin\theta=\dfrac{\text{perpendicular}}{\text{hypotenuse}},\quad \cos\theta=\dfrac{\text{base}}{\text{hypotenuse}},\quad \tan\theta=\dfrac{\text{perpendicular}}{\text{base}}
Variables
θ\theta = acute angle of a right triangle.
Conditions
Valid for acute angles in right triangle definition.
Where used in JEE
Basic evaluation of trigonometric ratios from geometry.
Remaining trigonometric ratiosEssential
csc⁡θ=1sin⁡θ=hypotenuseperpendicular,sec⁡θ=1cos⁡θ=hypotenusebase,cot⁡θ=1tan⁡θ=baseperpendicular\csc\theta=\dfrac{1}{\sin\theta}=\dfrac{\text{hypotenuse}}{\text{perpendicular}},\quad \sec\theta=\dfrac{1}{\cos\theta}=\dfrac{\text{hypotenuse}}{\text{base}},\quad \cot\theta=\dfrac{1}{\tan\theta}=\dfrac{\text{base}}{\text{perpendicular}}
Variables
θ\theta = acute angle of a right triangle.
Conditions
Denominators nonzero.
Where used in JEE
Reciprocal and quotient transformations.
Coordinate definition of trigonometric functionsImportant
sin⁡θ=yr,cos⁡θ=xr,tan⁡θ=yx\sin\theta=\dfrac{y}{r},\quad \cos\theta=\dfrac{x}{r},\quad \tan\theta=\dfrac{y}{x}
Variables
(x,y)(x,y) = point on terminal side of angle θ\theta, r=x2+y2r=\sqrt{x^2+y^2}.
Conditions
For tan⁡θ\tan\theta, x≠0x\neq 0.
Where used in JEE
Signs in quadrants, general-angle trigonometry.
Domain and range of sineEssential
sin⁡x∈[−1,1]\sin x\in[-1,1]
Variables
x∈Rx\in\mathbb{R}.
Where used in JEE
Solving equations and checking validity of expressions.
Domain and range of cosineEssential
cos⁡x∈[−1,1]\cos x\in[-1,1]
Variables
x∈Rx\in\mathbb{R}.
Where used in JEE
Solving equations and checking validity of expressions.
Domain and range of tangent and cotangentEssential
tan⁡x∈R,cot⁡x∈R\tan x\in\mathbb{R},\quad \cot x\in\mathbb{R}
Variables
x∈Rx\in\mathbb{R}.
Conditions
tan⁡x\tan x undefined when x=(2n+1)π/2x=(2n+1)\pi/2; cot⁡x\cot x undefined when x=nπx=n\pi, n∈Zn\in\mathbb{Z}.
Where used in JEE
Domain restrictions in equations and transformations.
Domain and range of secant and cosecantImportant
sec⁡x∈(−∞,−1]∪[1,∞),csc⁡x∈(−∞,−1]∪[1,∞)\sec x\in(-\infty,-1]\cup[1,\infty),\quad \csc x\in(-\infty,-1]\cup[1,\infty)
Variables
x∈Rx\in\mathbb{R}.
Conditions
sec⁡x\sec x undefined when x=(2n+1)π/2x=(2n+1)\pi/2; csc⁡x\csc x undefined when x=nπx=n\pi, n∈Zn\in\mathbb{Z}.
Where used in JEE
Range-based equation solving and simplification.
Signs in quadrantsEssential
QI: all positive; QII: sin⁡,csc⁡>0; QIII: tan⁡,cot⁡>0; QIV: cos⁡,sec⁡>0\text{QI: all positive; QII: }\sin,\csc>0;\ \text{QIII: }\tan,\cot>0;\ \text{QIV: }\cos,\sec>0
Where used in JEE
Determining signs of trigonometric expressions in different quadrants.
Even-odd propertiesEssential
sin⁡(−x)=−sin⁡x,cos⁡(−x)=cos⁡x,tan⁡(−x)=−tan⁡x,cot⁡(−x)=−cot⁡x,sec⁡(−x)=sec⁡x,csc⁡(−x)=−csc⁡x\sin(-x)=-\sin x,\quad \cos(-x)=\cos x,\quad \tan(-x)=-\tan x,\quad \cot(-x)=-\cot x,\quad \sec(-x)=\sec x,\quad \csc(-x)=-\csc x
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
Simplification, proving identities, solving equations.
Periodicity of trigonometric functionsEssential
sin⁡(x+2π)=sin⁡x,cos⁡(x+2π)=cos⁡x,sec⁡(x+2π)=sec⁡x,csc⁡(x+2π)=csc⁡x,tan⁡(x+π)=tan⁡x,cot⁡(x+π)=cot⁡x\sin(x+2\pi)=\sin x,\quad \cos(x+2\pi)=\cos x,\quad \sec(x+2\pi)=\sec x,\quad \csc(x+2\pi)=\csc x,\quad \tan(x+\pi)=\tan x,\quad \cot(x+\pi)=\cot x
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
General solutions and transformation of angles.
Cofunction identitiesEssential
sin⁡(π2−x)=cos⁡x,cos⁡(π2−x)=sin⁡x,tan⁡(π2−x)=cot⁡x,cot⁡(π2−x)=tan⁡x,sec⁡(π2−x)=csc⁡x,csc⁡(π2−x)=sec⁡x\sin\left(\dfrac{\pi}{2}-x\right)=\cos x,\quad \cos\left(\dfrac{\pi}{2}-x\right)=\sin x,\quad \tan\left(\dfrac{\pi}{2}-x\right)=\cot x,\quad \cot\left(\dfrac{\pi}{2}-x\right)=\tan x,\quad \sec\left(\dfrac{\pi}{2}-x\right)=\csc x,\quad \csc\left(\dfrac{\pi}{2}-x\right)=\sec x
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
Reduction and complementary-angle transformations.
Supplementary and allied angle identitiesEssential
sin⁡(π−x)=sin⁡x,cos⁡(π−x)=−cos⁡x,tan⁡(π−x)=−tan⁡x,sin⁡(π+x)=−sin⁡x,cos⁡(π+x)=−cos⁡x,tan⁡(π+x)=tan⁡x,sin⁡(2π−x)=−sin⁡x,cos⁡(2π−x)=cos⁡x,tan⁡(2π−x)=−tan⁡x\sin(\pi-x)=\sin x,\quad \cos(\pi-x)=-\cos x,\quad \tan(\pi-x)=-\tan x,\\ \sin(\pi+x)=-\sin x,\quad \cos(\pi+x)=-\cos x,\quad \tan(\pi+x)=\tan x,\\ \sin(2\pi-x)=-\sin x,\quad \cos(2\pi-x)=\cos x,\quad \tan(2\pi-x)=-\tan x
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
Reduction to acute angles and solving equations.
Allied angles with \(\pi/2\) shiftsImportant
sin⁡(π2+x)=cos⁡x,cos⁡(π2+x)=−sin⁡x,tan⁡(π2+x)=−cot⁡x,sin⁡(3π2+x)=−cos⁡x,cos⁡(3π2+x)=sin⁡x,tan⁡(3π2+x)=−cot⁡x\sin\left(\dfrac{\pi}{2}+x\right)=\cos x,\quad \cos\left(\dfrac{\pi}{2}+x\right)=-\sin x,\quad \tan\left(\dfrac{\pi}{2}+x\right)=-\cot x,\\ \sin\left(\dfrac{3\pi}{2}+x\right)=-\cos x,\quad \cos\left(\dfrac{3\pi}{2}+x\right)=\sin x,\quad \tan\left(\dfrac{3\pi}{2}+x\right)=-\cot x
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
Transformation and simplification of trigonometric expressions.
Reciprocal identitiesEssential
sin⁡x csc⁡x=1,cos⁡x sec⁡x=1,tan⁡x cot⁡x=1\sin x\,\csc x=1,\quad \cos x\,\sec x=1,\quad \tan x\,\cot x=1
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
Algebraic simplification and identity proofs.
Quotient identitiesEssential
tan⁡x=sin⁡xcos⁡x,cot⁡x=cos⁡xsin⁡x\tan x=\dfrac{\sin x}{\cos x},\quad \cot x=\dfrac{\cos x}{\sin x}
Variables
x∈Rx\in\mathbb{R}.
Conditions
Denominators nonzero.
Where used in JEE
Conversion between trigonometric ratios.
Pythagorean identitiesEssential
sin⁡2x+cos⁡2x=1,1+tan⁡2x=sec⁡2x,1+cot⁡2x=csc⁡2x\sin^2 x+\cos^2 x=1,\quad 1+\tan^2 x=\sec^2 x,\quad 1+\cot^2 x=\csc^2 x
Variables
x∈Rx\in\mathbb{R}.
Conditions
Second and third identities where defined.
Where used in JEE
Most trigonometric simplifications, equation solving, and substitutions.
Express one ratio in terms of anotherImportant
sin⁡2x=1−cos⁡2x,cos⁡2x=1−sin⁡2x,tan⁡2x=sec⁡2x−1,cot⁡2x=csc⁡2x−1\sin^2 x=1-\cos^2 x,\quad \cos^2 x=1-\sin^2 x,\quad \tan^2 x=\sec^2 x-1,\quad \cot^2 x=\csc^2 x-1
Variables
x∈Rx\in\mathbb{R}.
Conditions
Where defined.
Where used in JEE
Identity manipulation and reducing expressions.
Standard values at special anglesEssential
x0π/6π/4π/3π/2sin⁡x01/21/23/21cos⁡x13/21/21/20tan⁡x01/313undefined\begin{array}{c|ccccc}x&0&\pi/6&\pi/4&\pi/3&\pi/2\\ \hline \sin x&0&1/2&1/\sqrt2&\sqrt3/2&1\\ \cos x&1&\sqrt3/2&1/\sqrt2&1/2&0\\ \tan x&0&1/\sqrt3&1&\sqrt3&\text{undefined}\end{array}
Conditions
Use reciprocals for csc⁡x,sec⁡x,cot⁡x\csc x,\sec x,\cot x where defined.
Where used in JEE
Direct evaluation and simplification in objective questions.
General values at quadrantal anglesImportant
sin⁡nπ=0,cos⁡nπ=(−1)n,sin⁡((2n+1)π2)=(−1)n,cos⁡((2n+1)π2)=0\sin n\pi=0,\quad \cos n\pi=(-1)^n,\quad \sin\left(\dfrac{(2n+1)\pi}{2}\right)=(-1)^n,\quad \cos\left(\dfrac{(2n+1)\pi}{2}\right)=0
Variables
n∈Zn\in\mathbb{Z}.
Where used in JEE
Evaluation of trig functions at standard multiples of \(\pi\).
Angle sum and difference formulasEssential
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B,cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B,tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\sin(A\pm B)=\sin A\cos B\pm \cos A\sin B,\\ \cos(A\pm B)=\cos A\cos B\mp \sin A\sin B,\\ \tan(A\pm B)=\dfrac{\tan A\pm \tan B}{1\mp \tan A\tan B}
Variables
A,BA,B are angles.
Conditions
For tangent, denominator nonzero.
Where used in JEE
Expansion, reduction, solving equations, proving identities.
Cotangent sum and difference formulasImportant
cot⁡(A±B)=cot⁡Acot⁡B∓1cot⁡B±cot⁡A\cot(A\pm B)=\dfrac{\cot A\cot B\mp 1}{\cot B\pm \cot A}
Variables
A,BA,B are angles.
Conditions
Denominator nonzero; where defined.
Where used in JEE
Expression simplification and tangent-cotangent transformations.
Double angle formulasEssential
sin⁡2A=2sin⁡Acos⁡A,cos⁡2A=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A,tan⁡2A=2tan⁡A1−tan⁡2A,cot⁡2A=cot⁡2A−12cot⁡A\sin 2A=2\sin A\cos A,\\ \cos 2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A,\\ \tan 2A=\dfrac{2\tan A}{1-\tan^2A},\quad \cot 2A=\dfrac{\cot^2A-1}{2\cot A}
Variables
AA = angle.
Conditions
For tangent and cotangent, denominators nonzero.
Where used in JEE
Transformations, equation solving, maxima-minima style simplifications.
Triple angle formulasEssential
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^3A,\\ \cos 3A=4\cos^3A-3\cos A,\\ \tan 3A=\dfrac{3\tan A-\tan^3A}{1-3\tan^2A}
Variables
AA = angle.
Conditions
For tangent, denominator nonzero.
Where used in JEE
Equation solving, polynomial-trigonometric conversion.
Half-angle formulas for squaresEssential
sin⁡2A2=1−cos⁡A2,cos⁡2A2=1+cos⁡A2\sin^2\dfrac{A}{2}=\dfrac{1-\cos A}{2},\quad \cos^2\dfrac{A}{2}=\dfrac{1+\cos A}{2}
Variables
AA = angle.
Where used in JEE
Power reduction, integration-style simplification, equation solving.
Half-angle formulas for tangentEssential
tan⁡A2=±1−cos⁡A1+cos⁡A=sin⁡A1+cos⁡A=1−cos⁡Asin⁡A\tan\dfrac{A}{2}=\pm\sqrt{\dfrac{1-\cos A}{1+\cos A}}=\dfrac{\sin A}{1+\cos A}=\dfrac{1-\cos A}{\sin A}
Variables
AA = angle.
Conditions
Sign depends on quadrant of A/2A/2; denominators nonzero.
Where used in JEE
Substitution, simplification, proving identities.
Power reduction formulasEssential
sin⁡2x=1−cos⁡2x2,cos⁡2x=1+cos⁡2x2,sin⁡xcos⁡x=12sin⁡2x\sin^2x=\dfrac{1-\cos2x}{2},\quad \cos^2x=\dfrac{1+\cos2x}{2},\quad \sin x\cos x=\dfrac{1}{2}\sin2x
Variables
x∈Rx\in\mathbb{R}.
Where used in JEE
Reduction of powers and products.
Product to sum formulasEssential
sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)],cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)],sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)],cos⁡Asin⁡B=12[sin⁡(A+B)−sin⁡(A−B)]\sin A\sin B=\dfrac{1}{2}[\cos(A-B)-\cos(A+B)],\\ \cos A\cos B=\dfrac{1}{2}[\cos(A-B)+\cos(A+B)],\\ \sin A\cos B=\dfrac{1}{2}[\sin(A+B)+\sin(A-B)],\\ \cos A\sin B=\dfrac{1}{2}[\sin(A+B)-\sin(A-B)]
Variables
A,BA,B are angles.
Where used in JEE
Transformation of products, summation and equation problems.
Sum to product formulasEssential
sin⁡C+sin⁡D=2sin⁡C+D2cos⁡C−D2,sin⁡C−sin⁡D=2cos⁡C+D2sin⁡C−D2,cos⁡C+cos⁡D=2cos⁡C+D2cos⁡C−D2,cos⁡C−cos⁡D=−2sin⁡C+D2sin⁡C−D2\sin C+\sin D=2\sin\dfrac{C+D}{2}\cos\dfrac{C-D}{2},\\ \sin C-\sin D=2\cos\dfrac{C+D}{2}\sin\dfrac{C-D}{2},\\ \cos C+\cos D=2\cos\dfrac{C+D}{2}\cos\dfrac{C-D}{2},\\ \cos C-\cos D=-2\sin\dfrac{C+D}{2}\sin\dfrac{C-D}{2}
Variables
C,DC,D are angles.
Where used in JEE
Transformation of sums/differences, identity proof, equation solving.
General transformation to \(R\sin(x+\alpha)\)Essential
asin⁡x+bcos⁡x=Rsin⁡(x+α)a\sin x+b\cos x=R\sin(x+\alpha) where R=a2+b2, Rcos⁡α=a, Rsin⁡α=bR=\sqrt{a^2+b^2},\ R\cos\alpha=a,\ R\sin\alpha=b
Variables
a,b∈Ra,b\in\mathbb{R}, R>0R>0.
Conditions
α\alpha chosen according to signs of a,ba,b.
Where used in JEE
Finding maxima/minima, solving trigonometric equations, simplification.
General transformation to \(R\cos(x-\alpha)\)Essential
acos⁡x+bsin⁡x=Rcos⁡(x−α)a\cos x+b\sin x=R\cos(x-\alpha) where R=a2+b2, Rcos⁡α=a, Rsin⁡α=bR=\sqrt{a^2+b^2},\ R\cos\alpha=a,\ R\sin\alpha=b
Variables
a,b∈Ra,b\in\mathbb{R}, R>0R>0.
Conditions
α\alpha chosen according to signs of a,ba,b.
Where used in JEE
Amplitude form, extrema, equation solving.
Maximum and minimum of linear trig formEssential
−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}
Variables
a,b∈Ra,b\in\mathbb{R}, x∈Rx\in\mathbb{R}.
Where used in JEE
Range problems and optimization.
General solution of \(\sin x=\sin \alpha\)Essential
x=nπ+(−1)nα, n∈Zx=n\pi+(-1)^n\alpha,\ n\in\mathbb{Z}
Variables
α\alpha = given angle.
Where used in JEE
Solving standard trigonometric equations.
General solution of \(\cos x=\cos \alpha\)Essential
x=2nπ±α, n∈Zx=2n\pi\pm\alpha,\ n\in\mathbb{Z}
Variables
α\alpha = given angle.
Where used in JEE
Solving standard trigonometric equations.
General solution of \(\sec x=\sec \alpha\)Important
x=2nπ±α, n∈Zx=2n\pi\pm\alpha,\ n\in\mathbb{Z}
Variables
α\alpha = given angle.
Conditions
Equivalent to cos⁡x=cos⁡α\cos x=\cos\alpha.
Where used in JEE
Solving equations involving secant.
General solution of \(\csc x=\csc \alpha\)Important
x=nπ+(−1)nα, n∈Zx=n\pi+(-1)^n\alpha,\ n\in\mathbb{Z}
Variables
α\alpha = given angle.
Conditions
Equivalent to sin⁡x=sin⁡α\sin x=\sin\alpha.
Where used in JEE
Solving equations involving cosecant.
Formula using \(t=\tan(x/2)\)Important
sin⁡x=2t1+t2,cos⁡x=1−t21+t2,tan⁡x=2t1−t2\sin x=\dfrac{2t}{1+t^2},\quad \cos x=\dfrac{1-t^2}{1+t^2},\quad \tan x=\dfrac{2t}{1-t^2}
Variables
t=tan⁡(x/2)t=\tan(x/2).
Conditions
For tan⁡x\tan x, t≠±1t\neq \pm1.
Where used in JEE
Weierstrass substitution, equation solving, rationalization of trigonometric expressions.
Trigonometric ratios in terms of sides of a triangleImportant
asin⁡A=bsin⁡B=csin⁡C=2R\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R
Variables
a,b,ca,b,c are sides opposite angles A,B,CA,B,C; RR = circumradius.
Conditions
For any triangle ABCABC.
Where used in JEE
Sine rule applications in trigonometric triangles.
Cosine ruleImportant
a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A,\quad b2=c2+a2−2cacos⁡Bb^2=c^2+a^2-2ca\cos B,\quad c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C
Variables
a,b,ca,b,c are sides opposite angles A,B,CA,B,C.
Conditions
For any triangle.
Where used in JEE
Triangle solving and applications of trigonometry.
Projection formulas in a triangleSupplementary
a=bcos⁡C+ccos⁡B,b=ccos⁡A+acos⁡C,c=acos⁡B+bcos⁡Aa=b\cos C+c\cos B,\quad b=c\cos A+a\cos C,\quad c=a\cos B+b\cos A
Variables
a,b,ca,b,c are sides opposite angles A,B,CA,B,C.
Conditions
For any triangle.
Where used in JEE
Triangle transformations and relation derivations.
Area formulas of a triangle using trigonometryImportant
Δ=12bcsin⁡A=12casin⁡B=12absin⁡C\Delta=\dfrac{1}{2}bc\sin A=\dfrac{1}{2}ca\sin B=\dfrac{1}{2}ab\sin C
Variables
Δ\Delta = area of triangle, a,b,ca,b,c sides opposite angles A,B,CA,B,C.
Conditions
For any triangle.
Where used in JEE
Area-based triangle problems.
Tangent-half-angle in a triangleSupplementary
tan⁡A2=(s−b)(s−c)s(s−a)\tan\dfrac{A}{2}=\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}
Variables
a,b,ca,b,c are sides opposite angles A,B,CA,B,C; s=a+b+c2s=\dfrac{a+b+c}{2}.
Conditions
For a triangle.
Where used in JEE
Triangle geometry and half-angle evaluation.
Sine law with areaSupplementary
Δ=abc4R\Delta=\dfrac{abc}{4R}
Variables
Δ\Delta = area, a,b,ca,b,c = sides, RR = circumradius.
Conditions
For any triangle.
Where used in JEE
Mixed geometry-trigonometry problems.

Inverse trigonometrical functions and their properties

Principal value of inverse sineEssential
y=sin⁡−1x  ⟺  sin⁡y=x, y∈[−π2,π2]y=\sin^{-1}x\iff \sin y=x,\ y\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]
Variables
x∈[−1,1]x\in[-1,1].
Conditions
Range restricted to principal branch.
Where used in JEE
Evaluation, composition, equation solving with inverse trigonometric functions.
Principal value of inverse cosineEssential
y=cos⁡−1x  ⟺  cos⁡y=x, y∈[0,π]y=\cos^{-1}x\iff \cos y=x,\ y\in[0,\pi]
Variables
x∈[−1,1]x\in[-1,1].
Conditions
Range restricted to principal branch.
Where used in JEE
Evaluation, composition, equation solving with inverse trigonometric functions.
Principal value of inverse tangentEssential
y=tan⁡−1x  ⟺  tan⁡y=x, y∈(−π2,π2)y=\tan^{-1}x\iff \tan y=x,\ y\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)
Variables
x∈Rx\in\mathbb{R}.
Conditions
Principal branch excludes endpoints.
Where used in JEE
Evaluation, composition, and functional identities.
Principal value of inverse cotangentEssential
y=cot⁡−1x  ⟺  cot⁡y=x, y∈(0,π)y=\cot^{-1}x\iff \cot y=x,\ y\in(0,\pi)
Variables
x∈Rx\in\mathbb{R}.
Conditions
This is the standard JEE principal value range.
Where used in JEE
Evaluation and inverse trigonometric equation solving.
Principal value of inverse secantImportant
y=sec⁡−1x  ⟺  sec⁡y=x, y∈[0,π]∖{π2}y=\sec^{-1}x\iff \sec y=x,\ y\in[0,\pi]\setminus\left\{\dfrac{\pi}{2}\right\}
Variables
x∈(−∞,−1]∪[1,∞)x\in(-\infty,-1]\cup[1,\infty).
Conditions
Standard restricted range for principal value.
Where used in JEE
Evaluation and composition problems.
Principal value of inverse cosecantImportant
y=csc⁡−1x  ⟺  csc⁡y=x, y∈[−π2,π2]∖{0}y=\csc^{-1}x\iff \csc y=x,\ y\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\setminus\{0\}
Variables
x∈(−∞,−1]∪[1,∞)x\in(-\infty,-1]\cup[1,\infty).
Conditions
Standard restricted range for principal value.
Where used in JEE
Evaluation and composition problems.
Domains and ranges of inverse trigonometric functionsEssential
sin⁡−1x: D=[−1,1], R=[−π2,π2]cos⁡−1x: D=[−1,1], R=[0,π]tan⁡−1x: D=R, R=(−π2,π2)cot⁡−1x: D=R, R=(0,π)sec⁡−1x: D=(−∞,−1]∪[1,∞), R=[0,π]∖{π2}csc⁡−1x: D=(−∞,−1]∪[1,∞), R=[−π2,π2]∖{0}\begin{aligned}\sin^{-1}x:&\ D=[-1,1],\ R=\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\\ \cos^{-1}x:&\ D=[-1,1],\ R=[0,\pi]\\ \tan^{-1}x:&\ D=\mathbb{R},\ R=\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\\ \cot^{-1}x:&\ D=\mathbb{R},\ R=(0,\pi)\\ \sec^{-1}x:&\ D=(-\infty,-1]\cup[1,\infty),\ R=[0,\pi]\setminus\left\{\dfrac{\pi}{2}\right\}\\ \csc^{-1}x:&\ D=(-\infty,-1]\cup[1,\infty),\ R=\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\setminus\{0\}\end{aligned}
Where used in JEE
Checking validity of arguments and principal values.
Basic compositions with direct functionsEssential
sin⁡(sin⁡−1x)=x,cos⁡(cos⁡−1x)=x,tan⁡(tan⁡−1x)=x\sin(\sin^{-1}x)=x,\quad \cos(\cos^{-1}x)=x,\quad \tan(\tan^{-1}x)=x
Variables
xx in the domain of corresponding inverse function.
Where used in JEE
Simplification of compositions.
Reverse compositions with principal valuesEssential
sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x)=x for x∈[−π2,π2],cos⁡−1(cos⁡x)=xx\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right],\quad \cos^{-1}(\cos x)=x for x∈[0,π],tan⁡−1(tan⁡x)=xx\in[0,\pi],\quad \tan^{-1}(\tan x)=x for x∈(−π2,π2)x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)
Variables
x∈Rx\in\mathbb{R}.
Conditions
Only valid on principal value intervals.
Where used in JEE
Handling compositions carefully in JEE problems.
Expressions for \(\sin(\cos^{-1}x)\) and \(\cos(\sin^{-1}x)\)Essential
sin⁡(cos⁡−1x)=1−x2,cos⁡(sin⁡−1x)=1−x2\sin(\cos^{-1}x)=\sqrt{1-x^2},\quad \cos(\sin^{-1}x)=\sqrt{1-x^2}
Variables
x∈[−1,1]x\in[-1,1].
Conditions
Positive square root taken due to principal ranges.
Where used in JEE
Simplification of inverse-trig compositions.
Expressions for tangent forms from inverse sine and cosineImportant
tan⁡(sin⁡−1x)=x1−x2,tan⁡(cos⁡−1x)=1−x2x\tan(\sin^{-1}x)=\dfrac{x}{\sqrt{1-x^2}},\quad \tan(\cos^{-1}x)=\dfrac{\sqrt{1-x^2}}{x}
Variables
x∈[−1,1]x\in[-1,1].
Conditions
For second formula, x≠0x\neq 0; sign follows principal range of cos⁡−1x\cos^{-1}x.
Where used in JEE
Conversion of inverse-trig expressions to algebraic form.
Expressions for sine and cosine from inverse tangentEssential
sin⁡(tan⁡−1x)=x1+x2,cos⁡(tan⁡−1x)=11+x2\sin(\tan^{-1}x)=\dfrac{x}{\sqrt{1+x^2}},\quad \cos(\tan^{-1}x)=\dfrac{1}{\sqrt{1+x^2}}
Variables
x∈Rx\in\mathbb{R}.
Conditions
Cosine positive because tan⁡−1x∈(−π/2,π/2)\tan^{-1}x\in(-\pi/2,\pi/2).
Where used in JEE
Simplification and conversion to algebraic form.
Expressions involving inverse cotangentImportant
sin⁡(cot⁡−1x)=11+x2,cos⁡(cot⁡−1x)=x1+x2\sin(\cot^{-1}x)=\dfrac{1}{\sqrt{1+x^2}},\quad \cos(\cot^{-1}x)=\dfrac{x}{\sqrt{1+x^2}}
Variables
x∈Rx\in\mathbb{R}.
Conditions
Using principal range (0,π)(0,\pi); sign of cosine follows sign of xx.
Where used in JEE
Algebraic conversion of inverse cotangent expressions.
Relation between inverse tangent and inverse cotangentEssential
tan⁡−1x+cot⁡−1x=π2\tan^{-1}x+\cot^{-1}x=\dfrac{\pi}{2}
Variables
x∈Rx\in\mathbb{R}.
Conditions
With principal value range cot⁡−1x∈(0,π)\cot^{-1}x\in(0,\pi).
Where used in JEE
Simplification and evaluation of inverse-trig sums.
Relations for inverse secant and inverse cosecantSupplementary
sec⁡−1x+csc⁡−1x=π2sgn⁡(x)\sec^{-1}x+\csc^{-1}x=\dfrac{\pi}{2}\operatorname{sgn}(x)
Variables
x∈(−∞,−1]∪[1,∞)x\in(-\infty,-1]\cup[1,\infty), sgn⁡(x)=1\operatorname{sgn}(x)=1 for x>0x>0, −1-1 for x<0x<0.
Conditions
Using standard principal branches.
Where used in JEE
Advanced simplification of inverse reciprocal functions.
Negative argument propertiesEssential
sin⁡−1(−x)=−sin⁡−1x,cos⁡−1(−x)=π−cos⁡−1x,tan⁡−1(−x)=−tan⁡−1x,cot⁡−1(−x)=π−cot⁡−1x\sin^{-1}(-x)=-\sin^{-1}x,\quad \cos^{-1}(-x)=\pi-\cos^{-1}x,\quad \tan^{-1}(-x)=-\tan^{-1}x,\quad \cot^{-1}(-x)=\pi-\cot^{-1}x
Variables
xx in the domain of corresponding inverse function.
Conditions
Principal value conventions applied.
Where used in JEE
Simplification, symmetry, and equation solving.
Reciprocal inverse-function relationsImportant
sec⁡−1x=cos⁡−1(1x),csc⁡−1x=sin⁡−1(1x)\sec^{-1}x=\cos^{-1}\left(\dfrac{1}{x}\right),\quad \csc^{-1}x=\sin^{-1}\left(\dfrac{1}{x}\right)
Variables
∣x∣≥1|x|\ge 1.
Conditions
Principal values must be interpreted in standard ranges.
Where used in JEE
Converting reciprocal inverse functions to basic inverse functions.
Inverse tangent sum formulaEssential
tan⁡−1x+tan⁡−1y=tan⁡−1(x+y1−xy)\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\dfrac{x+y}{1-xy}\right)
Variables
x,y∈Rx,y\in\mathbb{R}.
Conditions
Principal value may require adding or subtracting π\pi depending on signs and value of xyxy; direct form valid when xy<1xy<1 and resultant principal value lies in range.
Where used in JEE
Evaluation of sums, especially in telescoping or numeric problems.
Inverse tangent difference formulaEssential
tan⁡−1x−tan⁡−1y=tan⁡−1(x−y1+xy)\tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\dfrac{x-y}{1+xy}\right)
Variables
x,y∈Rx,y\in\mathbb{R}.
Conditions
Principal value interpretation required.
Where used in JEE
Simplification and evaluation of inverse-trig differences.
Special case for inverse tangent sum to \(\pi/2\)Essential
tan⁡−1x+tan⁡−1(1x)={π2,x>0−π2,x<0\tan^{-1}x+\tan^{-1}\left(\dfrac{1}{x}\right)=\begin{cases}\dfrac{\pi}{2},&x>0\\-\dfrac{\pi}{2},&x<0\end{cases}
Variables
x∈R∖{0}x\in\mathbb{R}\setminus\{0\}.
Where used in JEE
Frequently tested inverse tangent evaluation.
Two-times inverse tangent identityImportant
2tan⁡−1x=tan⁡−1(2x1−x2)2\tan^{-1}x=\tan^{-1}\left(\dfrac{2x}{1-x^2}\right)
Variables
x∈Rx\in\mathbb{R}.
Conditions
Principal value may require adjustment by ±π\pm\pi when ∣x∣>1|x|>1.
Where used in JEE
Simplifying multiple-angle inverse tangent expressions.
Inverse sine sum formulaSupplementary
sin⁡−1x+sin⁡−1y=sin⁡−1(x1−y2+y1−x2)\sin^{-1}x+\sin^{-1}y=\sin^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right)
Variables
x,y∈[−1,1]x,y\in[-1,1].
Conditions
Valid when the sum lies in [−π/2,π/2][-\pi/2,\pi/2]; otherwise principal value adjustment required.
Where used in JEE
Advanced simplification and transformation of inverse-sine sums.
Inverse cosine sum relationSupplementary
cos⁡−1x+cos⁡−1y=cos⁡−1(xy−(1−x2)(1−y2))\cos^{-1}x+\cos^{-1}y=\cos^{-1}\left(xy-\sqrt{(1-x^2)(1-y^2)}\right)
Variables
x,y∈[−1,1]x,y\in[-1,1].
Conditions
Subject to principal value consistency.
Where used in JEE
Advanced inverse-cosine simplification.
Inverse sine and inverse cosine in terms of inverse tangentImportant
sin⁡−1x=tan⁡−1(x1−x2),cos⁡−1x=tan⁡−1(1−x2x)\sin^{-1}x=\tan^{-1}\left(\dfrac{x}{\sqrt{1-x^2}}\right),\quad \cos^{-1}x=\tan^{-1}\left(\dfrac{\sqrt{1-x^2}}{x}\right)
Variables
x∈[−1,1]x\in[-1,1].
Conditions
For second formula, principal value depends on sign of xx; if x<0x<0, adjustment by π\pi may be needed.
Where used in JEE
Interconversion among inverse trigonometric functions.
General principal value reduction for \(\sin^{-1}(\sin x)\)Important
sin⁡−1(sin⁡x)=(−1)n(x−nπ)\sin^{-1}(\sin x)=(-1)^n(x-n\pi) for x∈[nπ−π2, nπ+π2], n∈Zx\in\left[n\pi-\dfrac{\pi}{2},\ n\pi+\dfrac{\pi}{2}\right],\ n\in\mathbb{Z}
Variables
x∈Rx\in\mathbb{R}.
Conditions
Piecewise reduction to principal range of sin⁡−1\sin^{-1}.
Where used in JEE
Exact evaluation of inverse-trig compositions.
General principal value reduction for \(\cos^{-1}(\cos x)\)Important
cos⁡−1(cos⁡x)={x−2nπ,x∈[2nπ,(2n+1)π]2(n+1)π−x,x∈[(2n+1)π,2(n+1)π]\cos^{-1}(\cos x)=\begin{cases}x-2n\pi,&x\in[2n\pi,(2n+1)\pi]\\ 2(n+1)\pi-x,&x\in[(2n+1)\pi,2(n+1)\pi]\end{cases}
Variables
n∈Zn\in\mathbb{Z}.
Conditions
Output always in [0,π][0,\pi].
Where used in JEE
Exact evaluation of compositions with cosine inverse.
General principal value reduction for \(\tan^{-1}(\tan x)\)Important
tan⁡−1(tan⁡x)=x−nπ\tan^{-1}(\tan x)=x-n\pi where x∈(nπ−π2, nπ+π2), n∈Zx\in\left(n\pi-\dfrac{\pi}{2},\ n\pi+\dfrac{\pi}{2}\right),\ n\in\mathbb{Z}
Variables
x∈Rx\in\mathbb{R}.
Conditions
Output in (−π/2,π/2)(-\pi/2,\pi/2).
Where used in JEE
Exact evaluation and periodic adjustment problems.
Standard principal values of inverse trigonometric functionsImportant
sin⁡−10=0, sin⁡−11=π2, sin⁡−1(12)=π6, sin⁡−1(12)=π4, sin⁡−1(32)=π3\sin^{-1}0=0,\ \sin^{-1}1=\dfrac{\pi}{2},\ \sin^{-1}\left(\dfrac{1}{2}\right)=\dfrac{\pi}{6},\ \sin^{-1}\left(\dfrac{1}{\sqrt2}\right)=\dfrac{\pi}{4},\ \sin^{-1}\left(\dfrac{\sqrt3}{2}\right)=\dfrac{\pi}{3}
Conditions
Using principal values.
Where used in JEE
Quick evaluation in objective problems.
Standard principal values of inverse cosineImportant
cos⁡−11=0, cos⁡−1(32)=π6, cos⁡−1(12)=π4, cos⁡−1(12)=π3, cos⁡−10=π2, cos⁡−1(−1)=π\cos^{-1}1=0,\ \cos^{-1}\left(\dfrac{\sqrt3}{2}\right)=\dfrac{\pi}{6},\ \cos^{-1}\left(\dfrac{1}{\sqrt2}\right)=\dfrac{\pi}{4},\ \cos^{-1}\left(\dfrac{1}{2}\right)=\dfrac{\pi}{3},\ \cos^{-1}0=\dfrac{\pi}{2},\ \cos^{-1}(-1)=\pi
Conditions
Using principal values.
Where used in JEE
Quick evaluation in objective problems.
Standard principal values of inverse tangentImportant
tan⁡−10=0, tan⁡−1(13)=π6, tan⁡−11=π4, tan⁡−1(3)=π3\tan^{-1}0=0,\ \tan^{-1}\left(\dfrac{1}{\sqrt3}\right)=\dfrac{\pi}{6},\ \tan^{-1}1=\dfrac{\pi}{4},\ \tan^{-1}(\sqrt3)=\dfrac{\pi}{3}
Conditions
Using principal values.
Where used in JEE
Quick evaluation, inverse tangent sum problems.

Popular questions in Trigonometry

Frequently asked questions

What are the important Trigonometry formulas for JEE?

This page lists 80+ JEE-relevant Trigonometry formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Trigonometry formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Trigonometry, covering Trigonometrical identities and trigonometrical functions, Inverse trigonometrical functions and their properties.

How should I revise the Trigonometry formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Trigonometry MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Trigonometry?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.

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