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Geometry Spot/Formulas/Sum & Difference

Trigonometry Sum & Difference Formulas

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

Quick Reference
Essential Formulas
Basic Addition and Subtraction Formulas
The fundamental identities for sine, cosine, and tangent

Sine Formulas

Addition:
sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta
Subtraction:
sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta

Cosine Formulas

Addition:
cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta
Subtraction:
cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta

Tangent Formulas

Addition:
tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}
Subtraction:
tan⁡(α−β)=tan⁡α−tan⁡β1+tan⁡αtan⁡β\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}
Auxiliary Angle Formulas
Transform linear combinations into single trigonometric functions
General Form
asin⁡θ+bcos⁡θ=a2+b2sin⁡(θ+φ)a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \varphi)
where tan φ = b/a
Alternative Form
asin⁡θ+bcos⁡θ=a2+b2cos⁡(θ−ψ)a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\cos(\theta - \psi)
where tan ψ = a/b
Key Points:
  • R = √(a² + b²) gives the amplitude of the resulting function
  • The auxiliary angle φ depends on the quadrant of the point (a, b)
  • This transformation is useful for finding maximum/minimum values
Common Exact Values
Frequently used angles derived from addition/subtraction
AnglesincosDerivation
15°6−24\frac{\sqrt{6} - \sqrt{2}}{4}6+24\frac{\sqrt{6} + \sqrt{2}}{4}45° - 30°
75°6+24\frac{\sqrt{6} + \sqrt{2}}{4}6−24\frac{\sqrt{6} - \sqrt{2}}{4}45° + 30°
105°6+24\frac{\sqrt{6} + \sqrt{2}}{4}−6−24-\frac{\sqrt{6} - \sqrt{2}}{4}60° + 45°
165°6−24\frac{\sqrt{6} - \sqrt{2}}{4}−6+24-\frac{\sqrt{6} + \sqrt{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

Exact Value Calculation

sin 75° = sin(45° + 30°) = (√6 + √2)/4
cos 15° = cos(45° - 30°) = (√6 + √2)/4
tan 105° = tan(60° + 45°) = -2 - √3

Expression Simplification

sin 50° cos 10° + cos 50° sin 10° = sin 60° = √3/2
cos 35° cos 25° - sin 35° sin 25° = cos 60° = 1/2

Auxiliary Angle Applications

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