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Compound angle formulas (also called addition and subtraction formulas) are fundamental identities in trigonometry that express trigonometric functions of sums and differences of two angles in terms of the individual angles. These formulas are essential tools for simplifying trigonometric expressions, solving equations, and have extensive applications in physics, engineering, computer graphics, and signal processing. Understanding these identities unlocks the ability to evaluate trigonometric functions of non-standard angles and solve complex trigonometric problems.

The Six Compound Angle Identities

These formulas express the sine, cosine, and tangent of the sum or difference of two angles:

sin(A + B) = sin A cos B + cos A sin B

sin(A - B) = sin A cos B - cos A sin B

cos(A + B) = cos A cos B - sin 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)

tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Derivation Using the Unit Circle

The compound angle formulas can be derived geometrically using the unit circle. Consider two angles A and B with A greater than B greater than 0. Place points P and Q on the unit circle such that the angles measured from the positive x-axis are A and B respectively. Using the coordinates of these points and the distance formula, you can derive the sine and cosine addition formulas. The tangent formulas are then derived by dividing the sine formulas by the cosine formulas.

An intuitive way to remember the signs: for sin(A - B), think of it as sin(A + (-B)). Since cos(-B) = cos B and sin(-B) = -sin B, the formula naturally gives the minus sign.

Calculating Exact Values for Special Angles

One of the most practical applications of compound angle formulas is evaluating trigonometric functions of angles that aren't on the unit circle but can be expressed as sums or differences of special angles (30°, 45°, 60°).

sin(15°) = sin(45° - 30°) = sin45°cos30° - cos45°sin30°

= (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4

cos(15°) = cos(45° - 30°) = cos45°cos30° + sin45°sin30°

= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

tan(15°) = tan(45° - 30°) = (1 - 1/√3)/(1 + 1/√3) = 2 - √3

sin(75°) = sin(45° + 30°) = (√6 + √2)/4
cos(75°) = cos(45° + 30°) = (√6 - √2)/4
tan(75°) = tan(45° + 30°) = 2 + √3

Multiple Angle Formulas

Setting B = A in the compound angle formulas yields the double angle formulas:

sin(2A) = 2 sin A cos A

cos(2A) = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A

tan(2A) = 2 tan A / (1 - tan²A)

Example: If sin A = 3/5 and A is in first quadrant, find cos(2A).

cos A = √(1 - sin²A) = √(1 - 9/25) = 4/5

cos(2A) = cos²A - sin²A = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25

Product-to-Sum Identities

These identities convert products of trigonometric functions into sums or differences, which is particularly useful in integration:

2 sin A cos 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 Identities

The reverse of product-to-sum formulas:

sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)

sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2)

cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)

cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2)

Applications in Physics and Engineering

Wave interference: Adding two waves with phase differences
Alternating current: Analyzing AC circuits with phase angles
Signal processing: Modulation and demodulation of signals
Rotation matrices: 2D rotations using cos and sin of angle sums
Computer graphics: Rotating points and shapes
Navigation: Calculating bearings and directions

Problem-Solving Strategies

Look for angle combinations that match special angles (30°, 45°, 60°)
Use complementary angles: sin(90° - θ) = cos θ
For tan formulas, check that tan A tan B ≠ 1 (to avoid division by zero)
Double-check signs when subtracting angles
Convert all angles to the same format before applying formulas
When stuck, express everything in terms of sin and cos first

Compound Angle Calculator

Formulas

Compound Angle Formulas: Trigonometry Guide