Sum to Product identity of Sine functions

Formula

$\sin{\alpha}+\sin{\beta}$ $\,=\,$ $2\sin{\Big(\dfrac{\alpha+\beta}{2}\Big)}\cos{\Big(\dfrac{\alpha-\beta}{2}\Big)}$

An identity that expresses the transformation of sum of sine functions into product form is called the sum to product identity of sine functions.

Introduction

Let $\alpha$ and $\beta$ be two angles of right triangles. The sine functions with the two angles are written as $\sin{\alpha}$ and $\sin{\beta}$ mathematically. The sum of the two sine functions is written mathematically in the following form.

$\sin{\alpha}+\sin{\beta}$

The sum of sine functions can be transformed into the product of the trigonometric functions as follows.

$\implies$ $\sin{\alpha}+\sin{\beta}$ $\,=\,$ $2\sin{\Big(\dfrac{\alpha+\beta}{2}\Big)}\cos{\Big(\dfrac{\alpha-\beta}{2}\Big)}$

Other forms

The sum to product transformation rule of sin functions is popular written in two forms.

$(1). \,\,\,$ $\sin{x}+\sin{y}$ $\,=\,$ $2\sin{\Big(\dfrac{x+y}{2}\Big)}\cos{\Big(\dfrac{x-y}{2}\Big)}$

$(2). \,\,\,$ $\sin{C}+\sin{D}$ $\,=\,$ $2\sin{\Big(\dfrac{C+D}{2}\Big)}\cos{\Big(\dfrac{C-D}{2}\Big)}$

In the same way, you can write the sum to product transformation formula of sine functions in terms of any two angles.

Proof

Learn how to prove the sum to product transformation identity of sine functions in trigonometry.