Sin 2x Formula – Derivation, Values & Examples

The sin 2x formula is a double angle identity: $\sin(2x) = 2\sin(x)\cos(x)$.

Quick Reference:

Definition: The sine of twice an angle, expressed in terms of sin and cos of the original angle. Primary formula: $\sin(2x) = 2\sin(x)\cos(x)$ Alternative forms:

• $\sin(2x) = \dfrac{2\tan(x)}{1 + \tan^2(x)}$

• $\sin(2x) = \dfrac{2\tan(x)}{1 + \tan^2(x)}$ Type: Trigonometric double angle identity Used in: Trigonometry, calculus (integration), wave physics, signal processing

Full Definition

The sin 2x formula is one of the double angle identities — a set of equations that express trigonometric functions of $2x$ in terms of functions of $x$. The identity holds for all real values of $x$.

The formula is not an approximation — it is an exact algebraic identity derived from the angle addition formula.

Derivation of the sin 2x formula

Starting from the sine addition formula: $\sin(A + B) = \sin A \cos B + \cos A \sin B$.

Set $A = B = x$:

$$\sin(x + x) = \sin x \cos x + \cos x \sin x$$

$$\sin(2x) = 2\sin(x)\cos(x)$$

This is the primary form of the sin 2x formula — exact, simple, and derived in two steps.

An alternative form uses the identity $\cos^2(x) = \frac{1 - \tan^2(x)}{1 + \tan^2(x)}$ and $\sin(x)\cos(x) = \frac{\tan(x)}{1 + \tan^2(x)}$:

$$\sin(2x) = \frac{2\tan(x)}{1 + \tan^2(x)}$$

This form is useful when only $\tan(x)$ is known.

Values of sin 2x at Standard Angles

Note that $\sin(2x) = 1$ at $x = 45°$ — the maximum of the sine function is reached at $2x = 90°$.

Worked Examples

Example 1: Evaluating sin(2x) directly

Find $\sin(2x)$ when $x = 30°$.

$$\sin(2 \times 30°) = \sin(60°) = \frac{\sqrt{3}}{2}$$

Final answer: $\dfrac{\sqrt{3}}{2}$

Example 2: Using the formula

Find $\sin(2x)$ when $\sin(x) = \dfrac{3}{5}$ and $x$ is in the first quadrant.

First find $\cos(x)$ using $\sin^2(x) + \cos^2(x) = 1$: $$\cos(x) = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$$

Apply the formula: $$\sin(2x) = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}$$

Final answer: $\sin(2x) = \dfrac{24}{25}$

Common Confusions With The sin 2x Formula

$\sin(2x) \neq 2\sin(x)$. This is the most common error — multiplying the argument by 2 is not the same as multiplying the function value by 2. $\sin(2x) = 2\sin(x)\cos(x)$; the extra $\cos(x)$ factor is essential.

$\sin(2x) \neq \sin^2(x)$. These are entirely different expressions. $\sin^2(x) = \sin(x) \times \sin(x)$, while $\sin(2x) = 2\sin(x)\cos(x)$.

The period of $\sin(2x)$ is $180°$ (or $\pi$ radians), not $360°$. Doubling the argument halves the period.

Where The sin 2x Formula Appears

The sin 2x formula is used in integrating products of sin and cos: $\int \sin(x)\cos(x),dx = \frac{1}{2}\int \sin(2x),dx = -\frac{1}{4}\cos(2x) + C$. In physics, it appears in the equations for projectile range — the horizontal range of a projectile launched at angle $\theta$ is $R = \frac{v^2 \sin(2\theta)}{g}$, where the maximum range occurs at $\theta = 45°$ (giving $\sin(90°) = 1$). In signal processing, the double angle identity is used to describe harmonic generation — when a signal at frequency $f$ is doubled to $2f$.