Trigonometry formulas for JEE Main & Advanced

23 formula entries for Trigonometry, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Angle Measurement: Degrees & Radians

1^{\circ} = \frac{π}{180} rad

and

1 rad = \frac{180^{\circ}}{π}

When to use. Whenever angles are given in degrees or radians, especially in arc length, sector area, or trigonometric function evaluations.

Trap. Forgetting to convert degrees to radians when using calculus or arc length formulas.

Trigonometric Ratios: Definitions

\csc θ = \frac{1}{\sin θ}

\sec θ = \frac{1}{\cos θ}

\cot θ = \frac{1}{\tan θ}

When to use. For any problem involving right triangles or defining trigonometric functions in terms of sides.

Trap. Confusing opposite and adjacent sides for non-right triangles.

Pythagorean Identities

\sin^{2} θ + \cos^{2} θ = 1

1 + \tan^{2} θ = \sec^{2} θ

1 + \cot^{2} θ = \csc^{2} θ

When to use. To simplify expressions, prove identities, or solve equations involving squares of trig functions.

Trap. Forgetting the plus sign in $1 + \tan^{2} θ = \sec^{2} θ$ .

Sum and Difference Formulas

\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}

When to use. To find exact values of trig functions for angles like $15^{\circ}$ , $75^{\circ}$ , or to simplify expressions.

Trap. Mixing up signs: $\cos (A + B) = \cos A \cos B - \sin A \sin B$ (minus, not plus).

Double Angle Formulas

\sin 2 θ = 2 \sin θ \cos θ

\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ

\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}

When to use. To simplify expressions with $2 θ$ , solve equations, or integrate trig functions.

Trap. Forgetting the multiple forms of $\cos 2 θ$ ; each is useful in different contexts.

Half Angle Formulas

\sin \frac{θ}{2} = \pm \sqrt{\frac{1 - \cos θ}{2}}

\cos \frac{θ}{2} = \pm \sqrt{\frac{1 + \cos θ}{2}}

\tan \frac{θ}{2} = \pm \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = \frac{\sin θ}{1 + \cos θ} = \frac{1 - \cos θ}{\sin θ}

When to use. To find exact values for angles like ${22.5}^{\circ}$ , or to simplify integrals and equations.

Trap. Forgetting the $\pm$ sign; the sign depends on the quadrant of $θ / 2$ .

Product-to-Sum and Sum-to-Product Formulas

\sin A \cos B = \frac{1}{2} [\sin (A + B) + \sin (A - B)]

\cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)]

\sin A \sin B = \frac{1}{2} [\cos (A - B) - \cos (A + B)]

\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}

When to use. To simplify products of trig functions, solve equations, or integrate products.

Trap. Forgetting the factor $\frac{1}{2}$ in product-to-sum formulas.

Solving Trigonometric Equations

General solutions:

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

\cos θ = \cos α \Rightarrow θ = 2 n π \pm α

\tan θ = \tan α \Rightarrow θ = n π + α

, where

n \in Z

When to use. When given an equation involving trigonometric functions, find all angles satisfying it.

Trap. Forgetting to consider all quadrants when solving $\sin θ = k$ .

Trigonometric Ratios of Standard Angles

\sin 0 = 0

\sin 30^{\circ} = \frac{1}{2}

\sin 45^{\circ} = \frac{\sqrt{2}}{2}

\sin 60^{\circ} = \frac{\sqrt{3}}{2}

\sin 90^{\circ} = 1

; similarly for

\cos

and

\tan

When to use. For quick evaluation of trig functions at standard angles, especially in geometry and calculus.

Trap. Confusing $\sin 30^{\circ}$ with $\sin 60^{\circ}$ values.

Unit Circle and Sign of Trigonometric Functions

Quadrant signs: Q1: all +, Q2:

\sin

+, Q3:

\tan

+, Q4:

\cos

When to use. To determine sign of trig functions for any angle, especially in solving equations or evaluating expressions.

Trap. Forgetting that $\sec θ$ and $\csc θ$ have same sign as $\cos θ$ and $\sin θ$ respectively.

Graphs of Trigonometric Functions

y = \sin x

: domain

R

, range

[- 1, 1]

, period

2 π

;

y = \tan x

: domain

x \neq \frac{π}{2} + n π

, range

R

, period

π

When to use. To understand behavior of trig functions, solve inequalities, or find number of solutions.

Trap. Forgetting that $\tan x$ has vertical asymptotes, not defined at those points.

Inverse Trigonometric Functions: Principal Values

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

only if

x \in [- \frac{π}{2}, \frac{π}{2}]

; otherwise use periodicity.

When to use. To find angles from given trig values, especially in calculus and solving equations.

Trap. Assuming $\sin^{- 1} (\sin x) = x$ for all $x$ ; it's only true in the principal range.

Trigonometric Identities: Proving and Simplifying

Example:

\frac{\sin θ}{1 + \cos θ} + \frac{1 + \cos θ}{\sin θ} = 2 \csc θ

When to use. To prove trigonometric identities or simplify complex trig expressions.

Trap. Cross-multiplying without checking domain restrictions.

Trigonometric Ratios of Allied Angles

\sin (π - θ) = \sin θ

\cos (π - θ) = - \cos θ

\tan (π - θ) = - \tan θ

When to use. To evaluate trig functions of angles like $120^{\circ}$ , $135^{\circ}$ , etc., quickly.

Trap. Forgetting sign changes: $\cos (90^{\circ} + θ) = - \sin θ$ (negative).

Maximum and Minimum Values of Trigonometric Expressions

a \sin θ + b \cos θ = R \sin (θ + ϕ)

where

R = \sqrt{a^{2} + b^{2}}

\cos ϕ = a / R, \sin ϕ = b / R

When to use. To find max/min of expressions like $3 \sin x + 4 \cos x$ or to solve inequalities.

Trap. Forgetting the formula for $R$ ; it's $\sqrt{a^{2} + b^{2}}$ , not $a + b$ .

Sine and Cosine Laws for Triangles

Sine law:

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

; Cosine law:

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

When to use. To solve triangles (find unknown sides/angles) given some sides and angles.

Trap. Using sine law when the triangle is not acute; it still works but be careful with obtuse angles.

Area of a Triangle: Trigonometric Formula

Area = \frac{1}{2} a b \sin C = \frac{1}{2} b c \sin A = \frac{1}{2} c a \sin B

When to use. To find area when two sides and the included angle are known.

Trap. Using the wrong angle; it must be the angle between the two sides.

Periodicity and Symmetry of Trigonometric Functions

\sin (- θ) = - \sin θ

\cos (- θ) = \cos θ

\tan (- θ) = - \tan θ

When to use. To simplify expressions with negative angles or to find general solutions.

Trap. Assuming $\sin$ is even; it's odd.

Solving Trigonometric Inequalities

Example:

\sin x > \frac{1}{2}

gives

x \in (\frac{π}{6}, \frac{5 π}{6}) + 2 n π

When to use. When given an inequality involving trig functions, find the range of angles.

Trap. Forgetting to add $2 n π$ for general solutions.

Trigonometric Substitutions in Integration

\sqrt{a^{2} - x^{2}} = a \cos θ

when

x = a \sin θ

When to use. To integrate expressions involving square roots of quadratic forms.

Trap. Forgetting to change the differential $d x$ to $a \cos θ d θ$ .

Trigonometric Equations: Using Auxiliary Angle

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

with

\cos ϕ = a / R, \sin ϕ = b / R

When to use. To solve equations of the form $a \sin θ + b \cos θ = c$ .

Trap. Forgetting that $∣ c ∣ \leq R$ for solutions to exist.

Heights and Distances: Angle of Elevation/Depression

\tan θ = \frac{height}{distance}

When to use. In word problems involving towers, buildings, or objects at a distance.

Trap. Confusing angle of elevation with angle of depression; they are equal when lines are parallel.

Trigonometric Series: Summation of Sine/Cosine Series

Sum of

\sin

series:

\frac{\sin (n β / 2)}{\sin (β / 2)} \sin (α + (n - 1) β / 2)

When to use. To find sum of trigonometric series in arithmetic progression.

Trap. Forgetting the factor $\frac{\sin (n β / 2)}{\sin (β / 2)}$ when $β$ is not zero.

Values to remember

$π \approx 3.14159$ Angle Measurement: Degrees & Radians
$180^{\circ} = π rad$ Angle Measurement: Degrees & Radians
$\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ Sum and Difference Formulas
$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ Sum and Difference Formulas
$\sin 30^{\circ} = \frac{1}{2}$ Trigonometric Ratios of Standard Angles
$\cos 45^{\circ} = \frac{\sqrt{2}}{2}$ Trigonometric Ratios of Standard Angles
$\tan 60^{\circ} = \sqrt{3}$ Trigonometric Ratios of Standard Angles

Read the full chapter: Trigonometry for JEE, with what JEE asks and the common traps.