notesonly.in

One notebook for every subject — open it anywhere.

Log in

Product to Sum and Sum to Product Formulae Explained

MathematicsTry the interactive 3D model →

The product-to-sum formulae convert products of sines and cosines into sums, for example 2 sin A cos B = sin(A+B) + sin(A−B). The sum-to-product formulae run the other way, turning sin C + sin D into 2 sin((C+D)/2) cos((C−D)/2). Together they let you move freely between multiplied and added trigonometric forms.

They matter because calculus cannot easily integrate sin 50° cos 20°, but it integrates sin 70° + sin 30° without effort. In physics, adding two waves of nearly equal frequency produces beats, and the sum-to-product formula reveals the beat frequency directly.

Key Concepts

  • Derivation: add or subtract the sum and difference formulae; for example, sin(A+B) + sin(A−B) = 2 sin A cos B.
  • Four product forms: 2 sin A cos B, 2 cos A sin B, 2 cos A cos B and 2 sin A sin B each expand to a sum of single sines or cosines.
  • The odd one out: cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2) carries a leading minus sign; the other three do not.

Formulas

  • Product to sum: 2 sin A cos B = sin(A+B) + sin(A−B); 2 cos A sin B = sin(A+B) − sin(A−B); 2 cos A cos B = cos(A+B) + cos(A−B); 2 sin A sin B = cos(A−B) − cos(A+B).
  • Sum to product: sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2); sin C − sin D = 2 cos((C+D)/2) sin((C−D)/2); cos C + cos D = 2 cos((C+D)/2) cos((C−D)/2); cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2).
  • Assumptions: A, B, C, D are any real angles; no domain restrictions apply.

Worked Example

Express sin 75° − sin 15° as a product and evaluate it.

  1. Use sin C − sin D = 2 cos((C+D)/2) sin((C−D)/2) with C = 75°, D = 15°.
  2. (C+D)/2 = 90°/2 = 45° and (C−D)/2 = 60°/2 = 30°.
  3. So sin 75° − sin 15° = 2 cos 45° sin 30° = 2 × (√2/2) × (1/2) = √2/2.
  4. Numerical check: sin 75° ≈ 0.9659 and sin 15° ≈ 0.2588, giving 0.9659 − 0.2588 = 0.7071 = √2/2. The product form is exact and correct.

Common Misconceptions

  • The minus sign in cos C − cos D. It is the only sum-to-product formula with a leading negative; forgetting it flips the answer's sign.
  • 2 sin A sin B gives cosines, not sines: 2 sin A sin B = cos(A−B) − cos(A+B). Expecting a sine result is a common slip.
  • Direction confusion: product-to-sum expands, sum-to-product factors; applying the wrong direction leaves the expression unsimplified.

Real-World Applications

  • Integration: converting sin mx cos nx into a sum makes the integral straightforward.
  • Acoustics: beat frequencies from two close tones emerge from sin C + sin D factorization.
  • Radio engineering: amplitude modulation multiplies carrier and signal waves, analyzed with these formulae.
  • Solving equations: turning sin 5x + sin 3x = 0 into a product reveals the solutions immediately.

The product-to-sum and sum-to-product formulae translate between multiplied and added trigonometric forms, derived by adding and subtracting the sum and difference identities. Memorize all eight, watch the lone minus sign in cos C − cos D, and use the direction that simplifies your problem.

Related blogs