Sin 2x Formula – Derivation, Values & Examples

TL;DR

This article derives the sin 2x formula — sin 2x = 2 sin x cos x — from the angle addition identity, tabulates key values, and works through exam-style problems that use the double-angle identity to simplify expressions and solve equations. You will understand where the formula comes from and when to apply it.

The sin 2x formula is a double angle identity: sin(2x)=2sin(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)=2sin(x)cos(x)
Alternative forms:

• 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)=sinAcosB+cosAsinB.

Set A=B=x:

sin(x+x)=sin(x)cos(x)+cos(x)sin(x)

sin(2x)=2sin(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 cos²(x)=\frac{1 - \tan²(x)}{1 + \tan²(x)} and sin(x)cos(x)=\frac{\tan(x)}{1 + \tan²(x)}:

sin(2x)=\dfrac{2\tan(x)}{1 + \tan²(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(2×30°)=sin(60°)=\dfrac{\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 sin²(x)+cos²(x)=1: cos(x)=\sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}.

Apply the formula: sin(2x)=2×\frac{3}{5}×\frac{4}{5} = \dfrac{24}{25}.

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

Common Confusions With The sin 2x Formula

sin(2x)≠2sin(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)=2sin(x)cos(x); the extra cos(x) factor is essential.

sin(2x)≠sin²(x). These are entirely different expressions. sin²(x)=sin(x)×sin(x), while sin(2x)=2sin(x)cos(x).

The period of sin(2x) is 180° (or π 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: ∫sin(x)cos(x),dx=\frac{1}{2}∫sin(2x),dx. In physics, it appears in the equations for projectile range — the horizontal range of a projectile launched at angle θ is R=\frac{v² sin(2θ)}{g}, where the maximum range occurs at θ=45°. In signal processing, the double angle identity is used to describe harmonic generation.

Frequently Asked Questions

What is the sin 2x formula?
The sin 2x formula is sin(2x)=2sin(x)cos(x). It is a double angle identity derived from the sine addition formula by setting both angles equal to x.

Is sin(2x) the same as 2sin(x)?
No. sin(2x)=2sin(x)cos(x). The factor of cos(x) is not present in 2sin(x). The two expressions are only equal when cos(x)=1.

Where is the sin 2x formula used in calculus?
In integration: ∫sin(x)cos(x),dx is simplified using sin(x)cos(x)=\frac{1}{2}sin(2x).