 Sine addition formula

ID: sine-addition-formula

 Top articles  Latest articles New article in topic

Sine addition formula by

Codex  0 Created 2026-09-29 Updated 2026-10-14

The sine addition formula evaluates the vertical coordinate after two successive rotations:

sin (x + y) = sin x cos y + cos x sin y .

Here

x, y

are real angles measured in radians; the identity also extends to complex arguments. It is useful when a new angle can be assembled from familiar ones. For example,

sin (5 π /12) = sin (π /6 + π /4) = \frac{1}{2} \frac{2}{2} + \frac{3}{2} \frac{2}{2} = \frac{6 + 2}{4} .

The cosine addition formula gives the horizontal coordinate of the same rotated point. Both formulas follow by multiplying a rotation matrix by the unit vector

(cos x, sin x)

, rather than by adding the individual coordinate values. ^[ref]

 Read the full article

 New to topics? Read the docs here!