Trigonometry Sum and Difference Formulas

Complete reference for trigonometric addition and subtraction identities, including auxiliary angle transformations and common exact values.

Basic Addition and Subtraction Formulas

The fundamental identities for sine, cosine, and tangent

Addition:sin(α + β) = sin α cos β + cos α sin β

Subtraction:sin(α - β) = sin α cos β - cos α sin β

Addition:cos(α + β) = cos α cos β - sin α sin β

Subtraction:cos(α - β) = cos α cos β + sin α sin β

Addition:tan(α + β) = (tan α + tan β)/(1 - tan α tan β)

Subtraction:tan(α - β) = (tan α - tan β)/(1 + tan α tan β)

Auxiliary Angle Formulas

Transform linear combinations into single trigonometric functions

General Form

a sin θ + b cos θ = √(a² + b²) sin(θ + φ)

where tan φ = b/a

Alternative Form

a sin θ + b cos θ = √(a² + b²) cos(θ - ψ)

where tan ψ = a/b

Key Points:

Common Exact Values

Frequently used angles derived from addition/subtraction

Angle	sin	cos	Derivation
15°	(√6 - √2)/4	(√6 + √2)/4	45° - 30°
75°	(√6 + √2)/4	(√6 - √2)/4	45° + 30°
105°	(√6 + √2)/4	-(√6 - √2)/4	60° + 45°
165°	(√6 - √2)/4	-(√6 + √2)/4	180° - 15°

Memory Aids

Techniques for remembering the formulas

Sine Addition/Subtraction

sin cos + cos sin / sin cos - cos sin

Sine positive, cosine-sine pairs

Cosine Addition/Subtraction

cos cos - sin sin / cos cos + sin sin

Cosine negative (for +), sine-sine pairs

Tangent Addition/Subtraction

(tan + tan)/(1 - tan×tan) / (tan - tan)/(1 + tan×tan)

Fraction form: sum or difference over (1 ∓ product)

Applications and Examples

Common uses of sum and difference identities

sin 75° = sin(45° + 30°) = (√6 + √2)/4

cos 15° = cos(45° - 30°) = (√6 + √2)/4

tan 105° = tan(60° + 45°) = -2 - √3

sin 50° cos 10° + cos 50° sin 10° = sin 60° = √3/2

cos 35° cos 25° - sin 35° sin 25° = cos 60° = 1/2

sin x + cos x = √2 sin(x + π/4) → max value = √2

3 sin x + 4 cos x = 5 sin(x + φ) where tan φ = 4/3