Math Doubts

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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