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Sum Formulas in Trigonometry

Definition

Sum formulas (also called angle-addition formulas) allow exact evaluation of sine, cosine, and tangent at an angle that is the sum of two other known angles (e.g., 75° = 45° + 30°).

Why It Matters

These formulas allow for the exact evaluation of complex rotations without relying on calculators. They are the starting point for deriving the derivatives of sine and cosine and are essential for wave superposition analysis in signal processing or quantum mechanics.

Core Concepts

  • Sine of sum: 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 plus B equals sine-A cosine-B plus cosine-A sine-B.”
    • Meaning / mnemonic / when to use: The arguments are “mixed”. The sign is the same as the sum operation (+).
  • Cosine of sum: 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 plus B equals cosine-A cosine-B minus sine-A sine-B.”
    • Meaning / mnemonic / when to use: The arguments are “grouped”. The sign is the opposite of the sum operation (−). Essential for rotation matrices.
  • Tangent of sum: 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 plus B equals (tan A plus tan B) divided by (1 minus tan A tan B).”
    • Meaning / when to use: The numerator follows the sum sign, denominator is opposite. Use for angle sums given slopes.

Connected Concepts

Connected notes