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Product-to-Sum Formulas

Definition

These identities allow for the conversion between products of trigonometric functions and their sums or differences. They are primarily used in calculus and signal processing to simplify complex waveforms.

  • sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]
  • sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]
  • cos⁡Asin⁡B=12[sin⁡(A+B)−sin⁡(A−B)]\cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)]
  • cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]

Why It Matters

These identities are the difference between an unsolvable integral and a trivial one. In signal processing, failing to recognize these relationships leads to an inability to separate mixed frequencies (beats) or optimize bandwidth, causing “noise” in both mathematical models and physical communication systems.

Core Concepts

  • Product-to-Sum identities (turn products into sums — great for integration)

    • sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]

      • How to read: “The sine A cosine B equals one-half times the quantity sine of A plus B plus sine of A minus B.”
      • Meaning: Converts a sine-cosine product into a sum—useful for integrating products of trig functions.

    • sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]

      • How to read: “The sine A sine B equals one-half times the quantity cosine of A minus B minus cosine of A plus B.”
      • Meaning: Converts a sine-sine product into a cosine difference—simplifies integration and beat analysis.

    • cos⁡Asin⁡B=12[sin⁡(A+B)−sin⁡(A−B)]\cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)]

      • How to read: “The cosine A sine B equals one-half times the quantity sine of A plus B minus sine of A minus B.”
      • Meaning: Converts a cosine-sine product into a sine difference—mirror of the sine-cosine identity.

    • cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]

      • How to read: “The cosine A cosine B equals one-half times the quantity cosine of A minus B plus cosine of A plus B.”
      • Meaning: Converts a cosine-cosine product into a cosine sum—reveals sum and difference frequency components.

  • When to use: Use product-to-sum when you must integrate a product of two different trig functions (the integral of a sum is easy).

Connected Concepts

Connected notes