JEE Trigonometry All Formulas

Preparing for the Joint Entrance Examination (JEE) requires a thorough understanding of several mathematical concepts, and trigonometry is one of the most essential topics. Trigonometry plays a crucial role in both JEE Main and JEE Advanced. Questions from this topic often appear in a variety of formats, including direct formula-based problems and complex applications. To excel in JEE mathematics, mastering all trigonometric formulas and their applications is not optional it is necessary. Below is a detailed compilation of all the trigonometry formulas that are important for JEE, explained in a clear and organized way for easier learning and revision.

Basic Trigonometric Ratios

Trigonometric Ratios of Angles

The six fundamental trigonometric ratios are:

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent = sin(θ)/cos(θ)
• cot(θ) = 1 / tan(θ)
• sec(θ) = 1 / cos(θ)
• cosec(θ) = 1 / sin(θ)

Reciprocal Identities

Each trigonometric function has a reciprocal identity:

• sin(θ) = 1 / cosec(θ)
• cos(θ) = 1 / sec(θ)
• tan(θ) = 1 / cot(θ)

Trigonometric Identities

Fundamental Identities

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = cosec²(θ)

Co-function Identities

These identities relate the trigonometric functions of complementary angles:

• sin(90° – θ) = cos(θ)
• cos(90° – θ) = sin(θ)
• tan(90° – θ) = cot(θ)
• cot(90° – θ) = tan(θ)
• sec(90° – θ) = cosec(θ)
• cosec(90° – θ) = sec(θ)

Negative Angle Identities

• sin(-θ) = -sin(θ)
• cos(-θ) = cos(θ)
• tan(-θ) = -tan(θ)
• cot(-θ) = -cot(θ)
• sec(-θ) = sec(θ)
• cosec(-θ) = -cosec(θ)

Trigonometric Formulas for Compound Angles

Addition and Subtraction Formulas

• sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
• cos(A ± B) = cos(A)cos(B) ∠sin(A)sin(B)
• tan(A ± B) = (tan A ± tan B) / (1 ∠tan A·tan B)

Double Angle Formulas

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) – sin²(θ) = 2cos²(θ) – 1 = 1 – 2sin²(θ)
• tan(2θ) = 2tan(θ) / (1 – tan²(θ))

Triple Angle Formulas

• sin(3θ) = 3sin(θ) – 4sin³(θ)
• cos(3θ) = 4cos³(θ) – 3cos(θ)
• tan(3θ) = (3tan(θ) – tan³(θ)) / (1 – 3tan²(θ))

Product to Sum and Sum to Product Formulas

Product to Sum

• sin(A)sin(B) = ½[cos(A – B) – cos(A + B)]
• cos(A)cos(B) = ½[cos(A – B) + cos(A + B)]
• sin(A)cos(B) = ½[sin(A + B) + sin(A – B)]

Sum to Product

• sin(A) ± sin(B) = 2sin[(A ± B)/2]cos[(A ∠B)/2]
• cos(A) + cos(B) = 2cos[(A + B)/2]cos[(A – B)/2]
• cos(A) – cos(B) = -2sin[(A + B)/2]sin[(A – B)/2]

Trigonometric Equations

General Solutions

Solutions of basic trigonometric equations:

• sin(θ) = 0 ⇒ θ = nπ
• cos(θ) = 0 ⇒ θ = (2n+1)π/2
• tan(θ) = 0 ⇒ θ = nπ

Wherenis an integer.

Trigonometry in Radians

Angle Conversion

• Degrees to Radians: (π / 180) à degrees
• Radians to Degrees: (180 / π) à radians

Trigonometric Values of Standard Angles

Values for 0°, 30°, 45°, 60°, and 90°

Angle	sin	cos	tan	cot	sec	cosec
0°	0	1	0	∞	1	∞
30°	1/2	√3/2	1/√3	√3	2/√3	2
45°	1/√2	1/√2	1	1	√2	√2
60°	√3/2	1/2	√3	1/√3	2	2/√3
90°	1	0	∞	0	∞	1

Inverse Trigonometric Formulas

Basic Identities

• sin⁻¹(x) + cos⁻¹(x) = π/2
• tan⁻¹(x) + cot⁻¹(x) = π/2
• sec⁻¹(x) + cosec⁻¹(x) = π/2

Inverse Functions

• sin(sin⁻¹(x)) = x
• cos(cos⁻¹(x)) = x
• tan(tan⁻¹(x)) = x

Applications in JEE Problems

Common Uses in Geometry and Calculus

Trigonometric identities and formulas often appear in questions involving:

• Heights and distances
• Coordinate geometry (circles, triangles)
• Limits and differentiation involving sin(x), tan(x), etc.
• Definite integrals with trigonometric substitution

Tips for Memorization

• Practice formulas regularly through problems
• Use flashcards for quick revision
• Group similar formulas together to find patterns

Mastering all JEE trigonometry formulas is not just about rote memorization. It’s about understanding when and how to use them in a problem-solving context. Trigonometry connects to various other parts of the syllabus, so a solid foundation will benefit you far beyond isolated questions. Whether it’s using double angle identities in integration or inverse functions in calculus, these formulas are essential tools in your JEE toolkit. Make sure to revise them regularly and apply them frequently through practice tests and mock exams for better retention and accuracy.