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Difference Formulas in Trigonometry

Definition

Difference formulas allow exact evaluation of sine, cosine, and tangent at an angle that is the difference of two known angles (e.g., 15° = 45° - 30°).

Why It Matters

They are essential for evaluating rotations, determining beat frequencies in wave analysis, and crucially, they yield the classic cofunction identities (by setting one angle to 90°) bridging sine and cosine.

Core Concepts

  • Sine of difference: sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B
    • How to read: “Sine of A minus B equals sine-A cosine-B minus cosine-A sine-B.”
    • Meaning / mnemonic / when to use: Arguments are “mixed”. Sign matches the operation (−).
  • Cosine of difference: cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A \sin B
    • How to read: “Cosine of A minus B equals cosine-A cosine-B plus sine-A sine-B.”
    • Meaning / mnemonic / when to use: Arguments are “grouped”. Sign is the opposite (+).
  • Tangent of difference: tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
    • How to read: “Tangent of A minus B equals (tan A minus tan B) divided by (1 plus tan A tan B).”
    • Meaning / when to use: Numerator matches the sign, denominator is opposite.
  • Cofunction Identities: Setting A = 90° yields the classic co-function relations: sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos \theta
    • How to read: “Sine of 90 degrees minus theta equals cosine of theta.”
    • Meaning: These are special cases of the difference formulas connecting complementary angles.

Connected Concepts

Connected notes