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Sum and Difference Formulae: sin(A±B), cos(A±B), tan(A±B)

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The sum and difference formulae expand the sine, cosine and tangent of a combined angle: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, and tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B). They are the engine room of trigonometry, generating the double-angle, triple-angle and product-to-sum formulae.

They matter because they produce exact values for angles like 75° and 15°, which no standard table lists. Splitting 75° into 45° + 30° and expanding gives a precise answer with square roots, no calculator required.

Key Concepts

  • Compound angle: an angle written as the sum or difference of two known angles, like 75° = 45° + 30°.
  • Sign pattern: sine keeps the sign of ±, cosine reverses it, and tangent uses opposite signs in numerator and denominator.
  • Not distributive: sin(A + B) is not sin A + sin B; the formula has four terms, not two.
  • Exact values: the formulae turn standard-angle values into exact values for new angles.

Formulas

  • Sine: sin(A + B) = sin A cos B + cos A sin B; sin(A − B) = sin A cos B − cos A sin B.
  • Cosine: cos(A + B) = cos A cos B − sin A sin B; cos(A − B) = cos A cos B + sin A sin B.
  • Tangent: tan(A + B) = (tan A + tan B)/(1 − tan A tan B); tan(A − B) = (tan A − tan B)/(1 + tan A tan B), valid where all tangents are defined and denominators are non-zero.
  • Assumptions: A and B are any real angles; for tangent, none of A, B, A ± B may be an odd multiple of 90°.

Worked Example

Find the exact value of sin 75°.

  1. Split the angle: 75° = 45° + 30°, both standard angles.
  2. Apply the sum formula: sin 75° = sin 45° cos 30° + cos 45° sin 30°.
  3. Substitute: (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4.
  4. Numerical check: √6 ≈ 2.4495 and √2 ≈ 1.4142, so (2.4495 + 1.4142)/4 = 3.8637/4 ≈ 0.9659, which matches the calculator value of sin 75°.
  5. The exact form (√6 + √2)/4 is the answer textbooks expect.

Common Misconceptions

  • sin(A + B) ≠ sin A + sin B. Test with A = B = 30°: sin 60° = √3/2 ≈ 0.866, but sin 30° + sin 30° = 1. They are not equal.
  • Cosine flips the sign. cos(A + B) uses a minus between the products, the opposite of what sine does.
  • The tangent denominator matters. Forgetting the 1 − tan A tan B term is the most common error in tan(A + B).

Real-World Applications

  • Exact values: computing sin 15°, cos 75° and tan 15° precisely for geometry and exams.
  • Wave physics: adding two waves of different phases uses sin(A + B) to find the resulting amplitude.
  • Signal processing: phase shifts in alternating current are handled with these formulae.
  • Solving equations: expanding sin(x + 30°) turns a shifted equation into a solvable one.

The sum and difference formulae expand sin(A ± B), cos(A ± B) and tan(A ± B) into products of single-angle ratios. Memorize the sign pattern (sine keeps, cosine flips, tangent splits across a fraction), split non-standard angles into standard ones, and exact values follow.

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