Sine double angle identity

Formula

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

A trigonometric identity that expresses the expansion of sine of double angle in sine and cosine of angle is called the sine of double angle identity.

Introduction

Let theta be an angle of a right triangle, the sine and cosine functions are written as $\sin{\theta}$ and $\cos{\theta}$ respectively. Similarly, the sine of double angle function is written as $\sin{(2\theta)}$ in mathematical form.

The sine double angle function can be expressed in sine and cosine of angle in product form as follows.

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

Usage

The sine of double angle identity is mainly used in two different cases in mathematics.

Expansion

It is used to expand the sine of double angle functions in sine and cosine functions.

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

Simplified form

It is used to simplify the product of sine and cosine functions as sine of double angle function.

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

Other forms

The angle in sine of double angle formula can be denoted by any symbol. So, the sine of double angle identity can be expressed in terms of any variable. It is also usually expressed in three other popular forms.

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

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

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

Proof

Learn how to derive the rule of sin double angle formula in trigonometry by geometric method.

Another form

The sin of double angle formula can also be expanded in terms of tan of angle mathematically.

$\sin{(2\theta)} \,=\, \dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$