Free Double Angle Formula Calculator

The double angle formula calculator provides a straightforward way to compute all the essential double angle identities—sine, cosine, and tangent—for any given angle. These identities are fundamental for proving other trigonometric relationships, simplifying complex expressions, and solving equations that involve double angles. This article defines what a double angle is, presents the core formulas, and explains how to get the most out of the calculator.

What Is a Double Angle?

A double angle means multiplying the original angle by two. For example:

• $90^\circ$ is the double of $45^\circ$.
• If the starting angle is $-\frac{\pi}{3}$, the double angle is $-\frac{2\pi}{3}$.

The double angle identities express the trigonometric functions of $2\theta$ in terms of functions of $\theta$.

Sin Double Angle Formula

The sine of a double angle is given by:

$$\sin(2\theta) = 2\sin\theta\cos\theta.$$

This result follows from the sine angle sum identity:

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta.$$

Setting $\alpha = \beta = \theta$ directly produces the sin double angle formula.

Cos Double Angle Formula

For cosine, three equivalent forms are commonly used:

$$\cos(2\theta) = \cos^2\theta - \sin^2\theta,$$ $$\cos(2\theta) = 2\cos^2\theta - 1,$$ $$\cos(2\theta) = 1 - 2\sin^2\theta.$$

The first equation originates from the cosine sum identity:

$$\cos(\alpha+\beta)=\cos\alpha\cos\beta - \sin\alpha\sin\beta,$$

which simplifies to $\cos(2\theta)=\cos^2\theta-\sin^2\theta$ when $\alpha=\beta=\theta$. The other two versions are derived by applying the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$ to replace either $\sin^2\theta$ or $\cos^2\theta$.

Tan Double Angle Formula

The tangent double angle identity is:

$$\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}.$$

It comes from the tangent sum formula:

$$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta},$$

and is obtained by letting $\alpha=\beta=\theta$.

How to Use the Double Angle Formula Calculator

The calculator is designed for quick and flexible use. Follow these steps:

1. Choose the angle unit – degrees or radians.
2. Input the angle value – for example, $\frac{\pi}{12}$ (radians) or $15^\circ$.
3. Toggle the step-by-step solution – shown by default; you can hide it if you only need the numeric result.
4. View the results – the tool instantly computes $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

Example with $\theta = \frac{\pi}{12}$ (15°)

• $\sin(2\theta) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$
• $\cos(2\theta) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$
• $\tan(2\theta) = \tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3}$

The calculator also shows the input angle in the complementary unit (degrees when radians are entered, and vice versa), making it easy to work across different measurement systems. Whether you are studying trigonometry, preparing for an exam, or solving a practical problem, the double angle formula calculator gives you the exact values and the reasoning behind them.