# Sine Double angle formula

The expansion of sine of double angle in terms of sine of angle and cosine of angle is called sine double angle formula. It is true of all the angles, So, it is also called as sine double angle identity.

For example, if angle is theta ($\theta$), then double angle is $2\theta$. The sine of double angle is written as $\sin{2\theta}$ and it is expanded as twice the product of sine of angle ($\sin{\theta}$) and cosine of angle ($\cos{\theta}$).

## Formula

$\sin{2\theta} \,=\, 2\sin{\theta}\cos{\theta}$

### Other form

The angle can be denoted by any letter but the expansion of the sine of double angle is same. Generally, sine of double angle usually appears as $\sin{2x}$, $\sin{2A}$ and $\sin{2\alpha}$ in mathematics.

$(1) \,\,\,\,\,\,$ $\sin{2x} \,=\, 2\sin{x}\cos{x}$

$(2) \,\,\,\,\,\,$ $\sin{2A} \,=\, 2\sin{A}\cos{A}$

$(3) \,\,\,\,\,\,$ $\sin{2\alpha} \,=\, 2\sin{\alpha}\cos{\alpha}$

#### Proof

Sine double angle rule can be derived mathematically in geometrical method on the basis of the relations between angle, double angle and sides of the triangle.