Angle Addition Formulas
Reference for sin(A±B), cos(A±B), and tan(A±B) angle addition formulas.
Covers unit circle derivation and double-angle and half-angle identity applications.
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
Key Notes
- The four core identities: sin(A+B) = sinA cosB + cosA sinB; sin(A−B) = sinA cosB − cosA sinB; cos(A+B) = cosA cosB − sinA sinB; cos(A−B) = cosA cosB + sinA sinB.
- Sign pattern for cosine: Notice the cosine addition formulas have the opposite sign in the middle — cos(A+B) subtracts, cos(A−B) adds. This is a common source of errors.
- Double-angle formulas are a special case: Setting B = A in the addition formulas gives sin(2A) = 2sinA cosA and cos(2A) = cos²A − sin²A directly.
- Exact values from combined angles: These formulas let you find exact trig values for angles like 75° = 45° + 30° or 15° = 45° − 30° without a calculator.
- Valid for all real angles: These are exact algebraic identities, not approximations. They hold for any angles A and B in radians or degrees.