= Sine addition formula
{title2=$\sin(x+y)=\sin x\cos y+\cos x\sin y$}
The sine addition formula evaluates the vertical coordinate after two successive <rotations>:
$$
\boxed{\sin(x+y)=\sin x\cos y+\cos x\sin y.}
$$
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,
$$
\sin(5\pi/12)=\sin(\pi/6+\pi/4)
=\frac12\frac{\sqrt2}{2}+\frac{\sqrt3}{2}\frac{\sqrt2}{2}
=\frac{\sqrt6+\sqrt2}{4}.
$$
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. https://dlmf.nist.gov/4.21.i[NIST DLMF, §4.21(i): trigonometric addition formulas]{ref}
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