Introduction
Sum and difference formulas allow you to compute the sine, cosine, and tangent of the sum or difference of two angles. These formulas are extremely useful for simplifying trigonometric expressions and solving equations.
Formulas
- Sine: sin(A ± B) = sin A cos B ± cos A sin B
- Cosine: cos(A ± B) = cos A cos B ∓ sin A sin B
- Tangent: tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
Note the ± and ∓ signs: the top formula uses the same sign, while the bottom formula uses the opposite sign.
Strategy for Using Formulas
- Identify the angles A and B in the problem.
- Determine whether it is a sum or difference situation (A + B or A − B).
- Choose the correct formula (sine, cosine, or tangent).
- Substitute known values and simplify carefully.
- Check results using approximate decimal values if necessary.
Examples
Example 1 — Sine of a Sum:
Find sin(75°) using sin(A + B) with A = 45° and B = 30°:
sin(75°) = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4
Example 2 — Cosine of a Difference:
Find cos(15°) using cos(A − B) with A = 45° and B = 30°:
cos(15°) = cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4
Example 3 — Tangent of a Sum:
Find tan(75°) using tan(A + B) with A = 45° and B = 30°:
tan(75°) = (tan 45° + tan 30°) / (1 − tan 45° tan 30°) = (1 + 1/√3) / (1 − 1·1/√3) = ( (√3 + 1)/√3 ) / ( (√3 − 1)/√3 ) = (√3 + 1)/(√3 − 1)
Practice Problems
- Compute sin(105°) using sum formula (A = 60°, B = 45°).
- Compute cos(120°) using difference formula (A = 150°, B = 30°).
- Compute tan(75°) using sum formula (A = 45°, B = 30°).
- Use sin(A − B) to simplify sin 80° − sin 50°.
- Verify that cos(A + B) = cos A cos B − sin A sin B for A = 60°, B = 30°.