# Sin double angle formula

### Expansion form

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

### Simplified form

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

### Introduction

It is called as sine double angle identity and used as a formula in two cases.

1. Sin of double angle is expanded as the product of twice of sin and cos of angle.
2. The product of twice of sin and cos of angle is simplified as sin of double angle.

#### How to use

Sine double angle identity is used to either expand or simplify the double angle functions like $\sin{2x}$, $\sin{2A}$, $\sin{2\alpha}$ and etc. For example,

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

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

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

#### Proof

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

### Alternate form

The sin double angle identity can be expanded in terms of tan of angle alternatively.

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

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