What is the most important trigonometric identity?

The Pythagorean identity sin²(θ) + cos²(θ) = 1 is fundamental and appears in many derivations. It comes directly from the unit circle and is the basis for many other identities. For more information, see resources such as Khan Academy trigonometric identities .

How do I remember the sum and difference formulas?

For sine: sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B) - the sign matches the operation. For cosine: cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B) - the sign is opposite. A mnemonic is "sine keeps the sign, cosine changes it."

When should I use double angle formulas?

Double angle formulas are useful when you need to simplify expressions involving 2θ, find exact values for angles like 15° (half of 30°), or solve equations where the angle is doubled. They also help in calculus when finding derivatives of trigonometric functions.

What is the difference between product-to-sum and sum-to-product formulas?

Product-to-sum formulas convert products like sin(A)sin(B) into sums or differences, which is useful for integration. Sum-to-product formulas do the reverse, converting sums like sin(A) + sin(B) into products, which helps in simplifying expressions and solving equations.

📐 Trigonometry Formulas

Complete trigonometry formulas reference: Pythagorean identities, sum and difference formulas, double angle, half angle, product-to-sum, sum-to-product, reduction formulas, and reciprocal identities.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

A comprehensive reference guide to all essential trigonometric formulas. This page includes basic definitions, Pythagorean identities, sum and difference formulas, double angle formulas, half angle formulas, product-to-sum and sum-to-product formulas, reduction formulas, and reciprocal identities. Perfect for students, engineers, and anyone working with trigonometry.

Basic Trigonometric Formulas

Fundamental definitions and relationships of trigonometric functions.

sin(θ) = opposite / hypotenuse

cos(θ) = adjacent / hypotenuse

tan(θ) = opposite / adjacent = sin(θ) / cos(θ)

cot(θ) = adjacent / opposite = cos(θ) / sin(θ)

sec(θ) = hypotenuse / adjacent = 1 / cos(θ)

csc(θ) = hypotenuse / opposite = 1 / sin(θ)

Pythagorean Identities

Fundamental identities derived from the Pythagorean theorem.

sin²(θ) + cos²(θ) = 1

1 + tan²(θ) = sec²(θ)

1 + cot²(θ) = csc²(θ)

Sum and Difference Formulas

Formulas for trigonometric functions of sum and difference of angles.

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

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

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

tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Double Angle Formulas

Formulas for trigonometric functions of double angles.

sin(2θ) = 2sin(θ)cos(θ)

cos(2θ) = cos²(θ) - sin²(θ)

cos(2θ) = 2cos²(θ) - 1

cos(2θ) = 1 - 2sin²(θ)

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

Half Angle Formulas

Formulas for trigonometric functions of half angles.

sin(θ/2) = ±√[(1 - cos(θ)) / 2]

cos(θ/2) = ±√[(1 + cos(θ)) / 2]

tan(θ/2) = sin(θ) / (1 + cos(θ)) = (1 - cos(θ)) / sin(θ)

Product-to-Sum Formulas

Formulas to convert products of trigonometric functions into sums.

sin(A)sin(B) = [cos(A - B) - cos(A + B)] / 2

cos(A)cos(B) = [cos(A - B) + cos(A + B)] / 2

sin(A)cos(B) = [sin(A + B) + sin(A - B)] / 2

Sum-to-Product Formulas

Formulas to convert sums of trigonometric functions into products.

sin(A) + sin(B) = 2sin[(A + B)/2]cos[(A - B)/2]

sin(A) - sin(B) = 2cos[(A + B)/2]sin[(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]

Reduction Formulas

Formulas for trigonometric functions of angles related to 90° and 180°.

sin(90° ± θ) = cos(θ)

sin(180° ± θ) = ∓sin(θ)

cos(90° ± θ) = ∓sin(θ)

cos(180° ± θ) = -cos(θ)

tan(90° ± θ) = -cot(θ)

tan(180° ± θ) = tan(θ)

Reciprocal Identities

Relationships between trigonometric functions and their reciprocals.

csc(θ) = 1 / sin(θ)

sec(θ) = 1 / cos(θ)

cot(θ) = 1 / tan(θ)

Quotient Identities

Relationships between trigonometric functions expressed as quotients.

tan(θ) = sin(θ) / cos(θ)

cot(θ) = cos(θ) / sin(θ)

Applications

Common use cases for trigonometric formulas.

Homework Problem Solving
Trigonometry Exams
Engineering Calculations
Physics Wave Problems

Frequently Asked Questions

Common questions about trigonometric formulas.

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