Trig Identities Calculator

Verify and explore trigonometric identities including Pythagorean, double angle, half angle, and more

Explore Trigonometric Identities

Identity Type

Angle (θ)

Basic Trigonometric Values for θ = 30°

sin(θ)

0.5

cos(θ)

0.866025

tan(θ)

0.57735

sec(θ)

1.154701

csc(θ)

cot(θ)

1.732051

Pythagorean Results

Main Identity

sin²(θ) + cos²(θ) = 1

Left side: 1 = Right side: 1✓ Verified

Alternative Forms

tan²(θ) + 1 = sec²(θ): 1.333333 = 1.333333

1 + cot²(θ) = csc²(θ): 4.000000 = 4.000000

Common Angle Examples

Complete Trigonometric Identities Reference

Pythagorean Identities

sin²(θ) + cos²(θ) = 1

tan²(θ) + 1 = sec²(θ)

1 + cot²(θ) = csc²(θ)

Double Angle Identities

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))

Half Angle Identities

sin(θ/2) = ±√((1 - cos(θ)) / 2)

cos(θ/2) = ±√((1 + cos(θ)) / 2)

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

Sum and Difference Identities

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))

Cofunction Identities

sin(θ) = cos(90° - θ) | cos(θ) = sin(90° - θ)

tan(θ) = cot(90° - θ) | cot(θ) = tan(90° - θ)

sec(θ) = csc(90° - θ) | csc(θ) = sec(90° - θ)

Identity Types

Pythagorean

Fundamental identity relating sin² and cos²

Double Angle

Functions of 2θ in terms of θ

Half Angle

Functions of θ/2 in terms of θ

Cofunction

Complementary angle relationships

Sum/Difference

Functions of (A ± B) combinations

Identity Tips

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Pythagorean identity is the most fundamental - memorize it first

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Double angle formulas help solve complex trigonometric equations

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Half angle formulas require attention to signs based on quadrants

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Cofunction identities show relationships between complementary angles

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Sum/difference identities are building blocks for other identities

Understanding Trigonometric Identities

What are Trigonometric Identities?

Trigonometric identities are mathematical equalities that involve trigonometric functions and are true for all values of the variables where both sides are defined. They are fundamental tools in trigonometry for simplifying expressions and solving equations.

Why are they Important?

•Simplify complex trigonometric expressions
•Solve trigonometric equations
•Calculate exact values for specific angles
•Essential for calculus and advanced mathematics

Applications

Engineering

Signal processing, wave analysis, and oscillation studies

Physics

Wave mechanics, periodic motion, and electromagnetic theory

Computer Graphics

3D transformations, rotations, and animation calculations

Memory Aids for Common Identities

Pythagorean Identity

Remember the unit circle: x² + y² = 1, where x = cos(θ) and y = sin(θ)

Double Angle

For sine: "2 times sine times cosine" (2sin(θ)cos(θ))

Sum Formulas

Sine: "sine cosine plus cosine sine" | Cosine: "cosine cosine minus sine sine"

Cofunction

"Co" means complementary: sin(θ) = cos(90° - θ)

Related Math Calculators

Double Angle Calculator

Calculate double angle formulas and identities

Half Angle Calculator

Calculate half angle formulas and identities

Sum & Difference Identities

Calculate sum and difference trigonometric identities