Trigonometry Formula Sheet — JEE Main Mathematics

Q: What are the most important Trigonometry formulas for JEE Main?

This Trigonometry formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Trigonometrical identities, Trigonometrical functions, Inverse trigonometrical functions and their properties — each shown with the key result and, where useful, a worked example.

Q: Is this Trigonometry formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

Q: How should I revise Trigonometry formulas?

Blurt the Trigonometry formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Every key Trigonometry formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 3 sub-topics

Trigonometrical identities
Trigonometrical functions
Inverse trigonometrical functions and their properties

Angles, Ratios & Fundamental Identities

Trigonometric functions: On the unit circle a point at angle

θ

(cos θ, sin θ)

, with

tan θ = \frac{sin θ}{cos θ}

and the reciprocals

csc, sec, cot

. Angles use radians:

π

rad

= 18 0^{\circ}

and an arc of length

l = r θ

The unit circle, the All-Sin-Tan-Cos quadrant signs, the fundamental identities, and a table of domains, ranges, periods and allied angles — The unit circle, quadrant signs, identities, ranges and periods.

★ Remember · The three Pythagorean identities

sin^{2} θ + cos^{2} θ = 1

1 + tan^{2} θ = sec^{2} θ

and

1 + cot^{2} θ = csc^{2} θ

. Almost every simplification starts by applying one of these to trade one ratio for another.

Signs, ranges & periods

▸Signs by quadrant: $A ll +$ , $S in +$ , $T an +$ , $C os +$ (I-IV).
▸ $sin, cos \in [- 1, 1]$ ; $sec, csc$ satisfy $∣ \cdot ∣ \geq 1$ ; $tan, cot \in R$ .
▸Period $2 π$ for $sin, cos, sec, csc$ ; period $π$ for $tan, cot$ .
▸ $cos (- θ) = cos θ$ (even); $sin (- θ) = - sin θ$ , $tan (- θ) = - tan θ$ (odd).

🎯 Exam · Allied angles in one line

For

9 0^{\circ} \pm θ

and

27 0^{\circ} \pm θ

the function

changes

(sin

\leftrightarrow

cos, tan

\leftrightarrow

cot); for

18 0^{\circ} \pm θ

and

36 0^{\circ} \pm θ

stays

. The sign comes from the quadrant (ASTC) of the original angle.

★ Remember · Max / min of a·sinθ + b·cosθ

a sin θ + b cos θ = a^{2} + b^{2} sin (θ + ϕ)

, so its range is

[- a^{2} + b^{2}, a^{2} + b^{2}]

. This single fact answers most 'find the maximum value' questions.

⚠️ Watch out · Radians vs degrees

The arc formula

l = r θ

and all calculus of trig functions need

θ

radians

. Mixing

90

(degrees) with

π /2

(radians) in the same expression is a frequent and costly slip.

🚫 Examiner Trap · Examiner traps

(1) ASTC: in QII only Sin

+

, QIII only Tan

+

, QIV only Cos

+

— used to fix the SIGN of allied angles. (2)

9 0^{\circ} \pm θ

and

27 0^{\circ} \pm θ

FLIP the ratio (sin

\leftrightarrow

cos, tan

\leftrightarrow

cot);

18 0^{\circ} \pm θ

36 0^{\circ} \pm θ

KEEP it. (3)

sin^{2} θ + cos^{2} θ = 1

but

1 + tan^{2} θ = sec^{2} θ

and

1 + cot^{2} θ = csc^{2} θ

— match the right pair. (4) Range:

∣ sin ∣, ∣ cos ∣ \leq 1

∣ sec ∣, ∣ csc ∣ \geq 1

;

a sin θ + b cos θ \in [- a^{2} + b^{2}, a^{2} + b^{2}]

Compound, Multiple & Sub-multiple Angles

Compound-angle formulas: The identities for

sin, cos, tan

A \pm B

in terms of the ratios of A and B. They generate every multiple-angle (

2 A, 3 A

) and half-angle (

A /2

) formula by suitable substitution.

★ Remember · Sum & difference

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) = \frac{tan A \pm tan B}{1 \mp tan A tan B}

. Note the

flipped sign

cos

and the denominator of

tan

Double & triple angles

▸ $sin 2 A = 2 sin A cos A = \frac{2 tan A}{1 + tan ^{2} A}$ .
▸ $cos 2 A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A = \frac{1 - tan ^{2} A}{1 + tan ^{2} A}$ .
▸ $tan 2 A = \frac{2 tan A}{1 - tan ^{2} A}$ .
▸ $sin 3 A = 3 sin A - 4 sin^{3} A$ ; $cos 3 A = 4 cos^{3} A - 3 cos A$ .

🎯 Exam · Power-reduction & half-angle

sin^{2} A = \frac{1 - cos 2 A}{2}

cos^{2} A = \frac{1 + cos 2 A}{2}

— used to integrate or to lower a power. With

t = tan \frac{A}{2}

sin A = \frac{2 t}{1 + t ^{2}}

cos A = \frac{1 - t ^{2}}{1 + t ^{2}}

★ Remember · Worth memorising

sin 1 5^{\circ} = \frac{3 - 1}{2 2}

cos 1 5^{\circ} = \frac{3 + 1}{2 2}

tan 1 5^{\circ} = 2 - 3

; and

tan 7 5^{\circ} = 2 + 3

. These appear constantly in objective problems.

⚠️ Watch out · cos2A has three forms — pick wisely

Choose the form of

cos 2 A

that matches what you have:

1 - 2 sin^{2} A

if you know

sin A

2 cos^{2} A - 1

if you know

cos A

, the

tan

form if you know

tan A

. The wrong choice creates needless work.

🚫 Examiner Trap · Examiner traps

(1) Mind the sign pairing:

cos (A + B) = cos A cos B - sin A sin B

(MINUS),

tan (A + B) = \frac{t a n A + t a n B}{1 - t a n A t a n B}

. (2)

cos 2 A

has THREE forms (

2 cos^{2} A - 1 = 1 - 2 sin^{2} A = cos^{2} A - sin^{2} A

) — pick the one matching what's given. (3) Half-angle

cos^{2} A = \frac{1 + c o s 2 A}{2}

(sign of

cos \frac{A}{2}

depends on the quadrant). (4) The

t = tan \frac{A}{2}

substitution gives

sin A = \frac{2 t}{1 + t ^{2}}

cos A = \frac{1 - t ^{2}}{1 + t ^{2}}

Transformation: Sum ↔ Product

Transformation formulas: Identities that convert a

sum

of two sines/cosines into a

product

(and back). They come straight from the compound-angle formulas and are the key to factoring and simplifying trig expressions.

★ Remember · Sum → product

sin C + sin D = 2 sin \frac{C + D}{2} cos \frac{C - D}{2}

;

cos C + cos D = 2 cos \frac{C + D}{2} cos \frac{C - D}{2}

;

cos C - cos D = - 2 sin \frac{C + D}{2} sin \frac{C - D}{2}

Product → sum

▸ $2 sin A cos B = sin (A + B) + sin (A - B)$ .
▸ $2 cos A sin B = sin (A + B) - sin (A - B)$ .
▸ $2 cos A cos B = cos (A + B) + cos (A - B)$ .
▸ $2 sin A sin B = cos (A - B) - cos (A + B)$ .

🎯 Exam · When A + B + C = π (triangle)

tan A + tan B + tan C = tan A tan B tan C

and

sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C

. These conditional identities turn up directly in triangle problems.

★ Remember · Direction of use

Going

sum \to product

helps factor (and solve equations, since a product

= 0

splits into cases); going

product \to sum

helps integrate and telescope series.

⚠️ Watch out · Mind the minus in cos C − cos D

cos C - cos D = - 2 sin \frac{C + D}{2} sin \frac{C - D}{2}

— the leading minus sign is easy to drop, and dropping it flips the sign of your whole answer.

🚫 Examiner Trap · Examiner traps

(1) Sum

\to

product to FACTOR/solve equations; product

\to

sum to INTEGRATE — choose by the goal. (2)

cos C - cos D = - 2 sin \frac{C + D}{2} sin \frac{C - D}{2}

— don't drop the leading MINUS. (3)

2 cos A cos B = cos (A + B) + cos (A - B)

but

2 sin A sin B = cos (A - B) - cos (A + B)

(order matters). (4) Conditional identities use

A + B + C = π

: e.g.

tan A + tan B + tan C = tan A tan B tan C

Trigonometric Equations

General solution: Because trig functions are periodic, an equation has infinitely many roots. The

general solution

captures them all using an integer n; the

principal solution

is the root(s) in

[0, 2 π)

General solution formulas for sin, cos and tan equations, a sine curve crossing a horizontal line at periodically spaced roots, and a method checklist — General solutions, the periodic roots of sin θ = k, and a method.

★ Remember · The three general solutions

sin θ = sin α \Rightarrow θ = nπ + (- 1)^{n} α

;

cos θ = cos α \Rightarrow θ = 2 nπ \pm α

;

tan θ = tan α \Rightarrow θ = nπ + α

, where

n \in Z

Special cases & method

▸ $sin θ = 0 \Rightarrow θ = nπ$ ; $cos θ = 0 \Rightarrow θ = (2 n + 1) \frac{π}{2}$ ; $tan θ = 0 \Rightarrow θ = nπ$ .
▸Reduce to $sin θ = sin α$ / $cos θ = cos α$ / $tan θ = tan α$ first.
▸ $sin θ = cos θ \Rightarrow tan θ = 1 \Rightarrow θ = nπ + \frac{π}{4}$ .
▸ $sin^{2} θ = sin^{2} α \Rightarrow θ = nπ \pm α$ (squared forms).

🎯 Exam · Reduce, solve, then place

Use identities to get a single function equal to a known value, apply the matching general solution, then list the principal solutions by giving

n

a few values. Always state the domain you were asked for.

⚠️ Watch out · Squaring/dividing creates or kills roots

Squaring

can introduce extraneous roots;

dividing

by a trig factor can drop the roots where that factor is zero. Substitute every candidate back, and never cancel

sin θ

cos θ

without considering

= 0

separately.

🚫 Examiner Trap · Examiner traps

(1)

sin θ = sin α \Rightarrow θ = nπ + (- 1)^{n} α

;

cos θ = cos α \Rightarrow θ = 2 nπ \pm α

;

tan θ = tan α \Rightarrow θ = nπ + α

— different forms,

n \in Z

. (2) SQUARING or dividing can introduce EXTRANEOUS roots / lose solutions — always check back. (3) Reduce to a single ratio

=

a standard value before applying the general solution. (4) Don't cancel a common factor without recording the roots it would give.

Properties of Triangles & Heights/Distances

Solving a triangle: In a triangle ABC the sides a,b,c face angles A,B,C. The

sine

and

cosine

rules relate the sides and angles so that, given enough data, every remaining side, angle, area and the circumradius R can be found.

A labelled triangle with the sine and cosine rules, area and half-angle formulas, and a small angle-of-elevation right triangle — A labelled triangle, the rules, and an elevation diagram.

★ Remember · Sine & cosine rules

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R

and

cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}

. Area

= \frac{1}{2} ab sin C = s (s - a) (s - b) (s - c)

with

s = \frac{a + b + c}{2}

Heights & distances

▸Angle of elevation: looking $up$ from the horizontal; depression: looking $down$ .
▸Model each sight-line as a right triangle and use $tan θ = \frac{opposite}{adjacent}$ .
▸Two observations give two equations — eliminate the unknown base distance.
▸Half-angle: $tan \frac{A}{2} = \frac{( s - b ) ( s - c )}{s ( s - a )}$ .

🎯 Exam · Which rule?

Use the

cosine

rule when you know two sides and the included angle, or all three sides; use the

sine

rule when you know an angle and its opposite side. The area form

\frac{1}{2} ab sin C

needs two sides and the included angle.

⚠️ Watch out · The ambiguous (SSA) case

Given two sides and a non-included angle, the sine rule can give

two

valid triangles (since

sin θ = sin (18 0^{\circ} - θ)

). Check both candidate angles against the triangle's angle sum before accepting them.

🚫 Examiner Trap · Examiner traps

(1) Sine rule

\frac{a}{s i n A} = 2 R

— side a is OPPOSITE angle A; the ambiguous (SSA) case can give TWO triangles. (2) Cosine rule

cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}

— note which side is isolated. (3) Area

= \frac{1}{2} ab sin C = s (s - a) (s - b) (s - c)

with

s = \frac{a + b + c}{2}

. (4) Heights & distances: angle of elevation is measured UP from horizontal, depression DOWN — set up

tan θ = \frac{opp}{adj}

carefully.

Inverse Trig — Definitions & Principal Values

Inverse trigonometric function:

y = sin^{- 1} x

means

sin y = x

AND y lies in the

principal-value branch

[- \frac{π}{2}, \frac{π}{2}]

. Each trig function is inverted only after restricting it to a branch on which it is one-to-one.

Graphs of arcsin, arccos and arctan, and a table of the domains and principal-value ranges of all six inverse trig functions — The inverse-trig graphs and the domain-to-range table.

★ Remember · Domains → principal ranges

sin^{- 1} : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}]

;

cos^{- 1} : [- 1, 1] \to [0, π]

;

tan^{- 1} : R \to (- \frac{π}{2}, \frac{π}{2})

;

cot^{- 1} : R \to (0, π)

Reciprocal branches & reading

▸ $sec^{- 1} : ∣ x ∣ \geq 1 \to [0, π] - {\frac{π}{2}}$ ; $csc^{- 1} : ∣ x ∣ \geq 1 \to [- \frac{π}{2}, \frac{π}{2}] - {0}$ .
▸Unstated branch always means the $principal value$ .
▸ $sin^{- 1} x$ is a function (single output), not the reciprocal $csc x$ .
▸Graphs of $sin^{- 1}$ and $cos^{- 1}$ stop at $x = \pm 1$ ; $tan^{- 1}$ runs over all x.

🎯 Exam · Always land in the branch

sin^{- 1} (sin θ) = θ

ONLY if

θ \in [- \frac{π}{2}, \frac{π}{2}]

, and

cos^{- 1} (cos θ) = θ

ONLY if

θ \in [0, π]

. Otherwise reduce the angle into the branch first using allied/periodic relations.

⚠️ Watch out · Inverse ≠ reciprocal

sin^{- 1} x

is the inverse function (an angle), while

(sin x)^{- 1} = csc x

is a reciprocal. The same '

- 1

' notation means completely different things — read it from the position.

🚫 Examiner Trap · Examiner traps

(1)

sin^{- 1} x

is the inverse FUNCTION, NOT

(sin x)^{- 1} = csc x

. (2) Always land in the PRINCIPAL range:

sin^{- 1} \in [- \frac{π}{2}, \frac{π}{2}]

cos^{- 1} \in [0, π]

tan^{- 1} \in (- \frac{π}{2}, \frac{π}{2})

. (3)

sin (sin^{- 1} x) = x

for

x \in [- 1, 1]

, but

sin^{- 1} (sin θ) = θ

ONLY if

θ \in [- \frac{π}{2}, \frac{π}{2}]

(else reduce first). (4) Domains:

sin^{- 1} / cos^{- 1}

need

∣ x ∣ \leq 1

;

sec^{- 1} / csc^{- 1}

need

∣ x ∣ \geq 1

Inverse Trig — Properties & Identities

Inverse-trig identities: A small set of relations — complementary, negative-argument, sum and double — that simplify almost every inverse-trig expression. Each result must be brought back into the relevant

principal range

★ Remember · Complementary & negative

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}

tan^{- 1} x + cot^{- 1} x = \frac{π}{2}

sec^{- 1} x + csc^{- 1} x = \frac{π}{2}

; and

sin^{- 1} (- x) = - sin^{- 1} x

cos^{- 1} (- x) = π - cos^{- 1} x

Sum & double-argument

▸ $tan^{- 1} x + tan^{- 1} y = tan^{- 1} \frac{x + y}{1 - x y}$ (for $x y < 1$ ).
▸ $tan^{- 1} x - tan^{- 1} y = tan^{- 1} \frac{x - y}{1 + x y}$ .
▸ $2 tan^{- 1} x = tan^{- 1} \frac{2 x}{1 - x ^{2}} = sin^{- 1} \frac{2 x}{1 + x ^{2}} = cos^{- 1} \frac{1 - x ^{2}}{1 + x ^{2}}$ .
▸ $sin^{- 1} x = cos^{- 1} 1 - x^{2} = tan^{- 1} \frac{x}{1 - x ^{2}}$ (for $x \geq 0$ ).

🎯 Exam · Inverse-then-direct collapses

sin (sin^{- 1} x) = x

for

x \in [- 1, 1]

, and

cos (sin^{- 1} x) = 1 - x^{2}

. Drawing a right triangle with the inner angle as

sin^{- 1} x

reads off any composite like

tan (cos^{- 1} x)

instantly.

⚠️ Watch out · The tan⁻¹ sum needs a quadrant check

tan^{- 1} x + tan^{- 1} y = tan^{- 1} \frac{x + y}{1 - x y}

holds only when

x y < 1

; if

x y > 1

add (or subtract)

π

so the answer stays in

(- \frac{π}{2}, \frac{π}{2})

. Always verify the sign and quadrant of the result.

🚫 Examiner Trap · Examiner traps

(1)

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}

(and the

tan^{- 1} / cot^{- 1}

sec^{- 1} / csc^{- 1}

pairs) — only on the valid domain. (2)

tan^{- 1} x + tan^{- 1} y = tan^{- 1} \frac{x + y}{1 - x y}

needs a

\pm π

correction when

x y > 1

(check the quadrant of the answer). (3)

2 tan^{- 1} x = tan^{- 1} \frac{2 x}{1 - x ^{2}} = sin^{- 1} \frac{2 x}{1 + x ^{2}}

hold only for

∣ x ∣ \leq 1

(or

∣ x ∣ < 1

). (4)

cos^{- 1} (cos θ) = θ

only for

θ \in [0, π]

; otherwise use

2 π - θ

- θ

More JEE Main Mathematics formula sheets

Frequently Asked Questions

What are the most important Trigonometry formulas for JEE Main?

This Trigonometry formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Trigonometrical identities, Trigonometrical functions, Inverse trigonometrical functions and their properties — each shown with the key result and, where useful, a worked example.

Is this Trigonometry formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Trigonometry formulas?

Blurt the Trigonometry formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheets · JEE Main previous-year papers · most important chapters.