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Sum-to-Product Formulas

Definition

These identities allow for the conversion between sums or differences of trigonometric functions and their products. They are primarily used to simplify waveforms and analyze beats.

  • sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
  • sin⁡A−sin⁡B=2cos⁡(A+B2)sin⁡(A−B2)\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)
  • cos⁡A+cos⁡B=2cos⁡(A+B2)cos⁡(A−B2)\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
  • cos⁡A−cos⁡B=−2sin⁡(A+B2)sin⁡(A−B2)\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)

Why It Matters

In signal processing, analyzing the sum of waves reveals phenomena like “beats.” Sum-to-product identities allow us to factor these sums into a single oscillating product, making it possible to find the carrier frequency and the envelope frequency.

Core Concepts

  • Sum-to-Product identities (turn sums into products — great for factoring and beats)

    • sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)

      • How to read: “The sine A plus sine B equals 2 sine of the quantity A plus B divided by 2 times cosine of the quantity A minus B divided by 2.”
      • Meaning: Factors a sine sum into product of half-angle-sum and half-angle-difference terms—key for solving trig equations and analyzing beats.

    • sin⁡A−sin⁡B=2cos⁡(A+B2)sin⁡(A−B2)\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)

      • How to read: “The sine A minus sine B equals 2 cosine of the quantity A plus B divided by 2 times sine of the quantity A minus B divided by 2.”
      • Meaning: Factors a sine difference—cosine of the average times sine of half the gap.

    • cos⁡A+cos⁡B=2cos⁡(A+B2)cos⁡(A−B2)\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)

      • How to read: “The cosine A plus cosine B equals 2 cosine of the quantity A plus B divided by 2 times cosine of the quantity A minus B divided by 2.”
      • Meaning: Factors a cosine sum into a product of cosines at half-sum and half-difference angles.

    • cos⁡A−cos⁡B=−2sin⁡(A+B2)sin⁡(A−B2)\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)

      • How to read: “The cosine A minus cosine B equals negative 2 sine of the quantity A plus B divided by 2 times sine of the quantity A minus B divided by 2.”
      • Meaning: Factors a cosine difference—negative product of sines at half-sum and half-difference angles.

  • When to use: Use sum-to-product when you see a sum or difference of sines/cosines that you want to factor (especially in solving trig equations or analyzing interference). The “average and half-difference” arguments are the key pattern to recognize.

Connected Concepts

Connected notes