Sine angle sum identity

Formula

$(1).\,\,$ $\sin{(a+b)}$ $\,=\,$ $\sin{a}\cos{b}$ $+$ $\cos{a}\sin{b}$

$(2).\,\,$ $\sin{(x+y)}$ $\,=\,$ $\sin{x}\cos{y}$ $+$ $\cos{x}\sin{y}$

Introduction

Let’s assume that $a$ and $b$ are two variables, which represent two different angles. The sum of angles is written as $a+b$, which is a compound angle. The sine of a compound angle $a$ plus $b$ is written as $\sin(a+b)$ in mathematics.

The sine of sum of angles $a$ and $b$ is equal to the sum of the product of sine of angle $a$ and cosine of angle $b$, and the product of cosine of angle $a$ and sine of angle $b$.

$\sin{(a+b)}$ $\,=\,$ $\sin{a} \times \cos{b}$ $+$ $\cos{a} \times \sin{b}$

This mathematical equation is called the sine angle sum trigonometric identity in mathematics.

Usage

The sine angle sum trigonometric identity is used in two different cases in mathematics.

Expansion

The sine of sum of two angles is expanded as the sum of products of sine and cosine of both angles.

$\implies$ $\sin{(a+b)}$ $\,=\,$ $\sin{(a)}\cos{(b)}$ $+$ $\cos{(a)}\sin{(b)}$

Simplification

The sum of products of sine and cosine of both angles is simplified as the sine of sum of two angles.

$\implies$ $\sin{(a)}\cos{(b)}$ $+$ $\cos{(a)}\sin{(b)}$ $\,=\,$ $\sin{(a+b)}$

Forms

The angle sum trigonometric identity in sine function is written in several forms but it is popularly expressed in the following three forms.

$(1).\,\,$ $\sin{(A+B)}$ $\,=\,$ $\sin{A}\cos{B}$ $+$ $\cos{A}\sin{B}$

$(2).\,\,$ $\sin{(x+y)}$ $\,=\,$ $\sin{x}\cos{y}$ $+$ $\cos{x}\sin{y}$

$(3).\,\,$ $\sin{(\alpha+\beta)}$ $\,=\,$ $\sin{\alpha}\cos{\beta}$ $+$ $\cos{\alpha}\sin{\beta}$

Proof

Learn how to derive the sine of angle sum trigonometric identity by a geometric approach in trigonometry.