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Sum and Difference Identities

Master the fundamental addition and subtraction formulas for trigonometric functions. These identities allow you to find exact values, simplify expressions, and solve complex trigonometric equations. The auxiliary angle formula provides a powerful tool for converting linear combinations of sine and cosine into a single sinusoidal function.

Core Addition and Subtraction Formulas
The six fundamental identities for sine, cosine, and tangent
Sine Addition
sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
Sine: positive, cosine-sine pairs
Sine Subtraction
sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
Sine: negative middle term
Cosine Addition
cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
Cosine: negative, sine-sine pair
Cosine Subtraction
cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
Cosine: positive, sine-sine pair
Tangent Addition
tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}
Fraction: sum over (1 minus product)
Tangent Subtraction
tan⁡(α−β)=tan⁡α−tan⁡β1+tan⁡αtan⁡β\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}
Fraction: difference over (1 plus product)
Auxiliary Angle Formula
Transform a sin α + b cos α into R sin(α + φ)
Formula: asin⁡α+bcos⁡α=a2+b2sin⁡(α+φ)a \sin \alpha + b \cos \alpha = \sqrt{a^2 + b^2} \sin(\alpha + \varphi)
where tan φ = b/a, and φ is determined by the signs of a and b

Example: 3sin α + 4cos α

• Calculate R = √(3² + 4²) = √25 = 5
• Find auxiliary angle: tan φ = b/a = 4/3
• Result: 3sin α + 4cos α = 5sin(α + φ)
where tan φ = 4/3, and φ is in the appropriate quadrant based on signs of a and b
Memory Aids
Techniques for remembering the formulas
  • Sine formulas: "Sine positive, cosine-sine pairs" - sin uses cos-sin combinations with + for addition
  • Cosine formulas: "Cosine negative, sine-sine pairs" - cos uses cos-cos first, then - sin-sin for addition
  • Tangent formulas: Fraction form with denominators involving (1 ∓ product)
  • Sign pattern: Addition formulas for sin and cos have opposite middle signs
Applications and Examples
Common uses of sum and difference identities
Exact Value Calculation
Calculate sin 75°, cos 15°, tan 105° using known angles
sin⁡75°=sin⁡(45°+30°)=6+24\sin 75° = \sin(45° + 30°) = \frac{\sqrt{6} + \sqrt{2}}{4}
Simplification
Simplify expressions like sin 30° cos 20° + cos 30° sin 20°
= sin(30° + 20°) = sin 50°
Maximum/Minimum Values
Find extrema of functions like a sin x + b cos x
sin⁡x+cos⁡x\sin x + \cos x has maximum value 2\sqrt{2}