# Sum to Product identity of Sin functions

## Formula

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

The transformation of sum of two sin functions into product form is called the sum to product identity of sin functions.

### Introduction

$\alpha$ and $\beta$ are two angles of two right triangles, the sin functions with the same angles are written as $\sin{\alpha}$ and $\sin{\beta}$ mathematically. In trigonometry, the sine functions are often involved in addition in both expressions and equations. The addition of sine functions are written as an expression as follows.

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

The sum of sin functions can be simplified possibly by transforming it into product form.

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

#### Popular 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 formula of sine functions in terms of any two angles.

#### Proof

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

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