Angle Addition Formulas
Calculate sine, cosine, and tangent of angle sums and differences.
The building blocks for many trig identities.
The Formula
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
The angle addition formulas express the trig functions of a sum or difference of two angles in terms of the individual angles. The double angle and half angle formulas are special cases of these.
Variables
| Symbol | Meaning |
|---|---|
| A, B | Any two angles |
| ± | Use + for addition, - for subtraction |
| ∓ | Opposite sign: - for addition, + for subtraction |
Example 1
Find the exact value of sin(75°) using sin(45° + 30°)
sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
sin(75°) = (√6 + √2)/4 ≈ 0.9659
Example 2
Find cos(15°) using cos(45° - 30°)
cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
cos(15°) = (√6 + √2)/4 ≈ 0.9659
When to Use It
Use the angle addition formulas when:
- Finding exact values of trig functions for non-standard angles
- Deriving other identities (double angle, half angle, product-to-sum)
- Simplifying expressions in calculus and physics
- Analyzing phase shifts in wave equations and signal processing