# Sine of sum of two angles

Sin of sum of two angles identity is a trigonometric formula and it is one of the most useful trigonometric identities in trigonometry. It is mainly used to expand when sin function contains a compound angle (in the form of sum of two angle) as angle.

## Formula

$\sin{(sum \, of \, two \, angles)}$ $\,=\,$ $\sin{(first \, angle)}\cos{(second \, angle)}$ $+$ $\cos{(first \, angle)}\sin{(second \, angle)}$

In order to avoid confusion, sine of sum of two angles formula is written in three ways internationally. You can follow one of them.

### $\sin{(A+B)}$ formula

$A$ and $B$ are two angles and sum of two angles is $A+B$, then $\sin{(A+B)}$ is expanded in terms sum of product of sine and cosine of angles $A$ and $B$ as follows.

$\sin{(A+B)}$ $\,=\,$ $\sin{A}\cos{B}$ $+$ $\cos{A}\sin{B}$

### $\sin{(x+y)}$ formula

$x$ and $y$ are two angles and sum of two angles is $x+y$, then $\sin{(x+y)}$ can be expanded in terms sum of product of sine and cosine of angles $x$ and $y$ in the following way.

$\sin{(x+y)}$ $\,=\,$ $\sin{x}\cos{y}$ $+$ $\cos{x}\sin{y}$

### $\sin{(\alpha+\beta)}$ formula

$\alpha$ and $\beta$ are two angles and sum of two angles is $\alpha+\beta$, then $\sin{(\alpha+\beta)}$ can be expanded in terms sum of product of sine and cosine of angles $\alpha$ and $\beta$ as written below.

$\sin{(\alpha+\beta)}$ $\,=\,$ $\sin{\alpha}\cos{\beta}$ $+$ $\cos{\alpha}\sin{\beta}$