Double Angle Formula Calculator

The double angle identities express trig functions of 2θ in terms of trig functions of θ alone. They show up constantly in calculus, signal processing, and physics — anywhere you need to simplify an expression involving a doubled angle.

sin(2θ) = 2 sin(θ) cos(θ) cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ) tan(2θ) = 2tan(θ) / (1 − tan²(θ))

The cosine formula has three equivalent forms. Which one you use depends on what you are already working with. If you only know sin(θ), use 1 − 2sin²(θ). If you only know cos(θ), use 2cos²(θ) − 1. The third form, cos²(θ) − sin²(θ), is the starting point for deriving the other two.

Tan(2θ) is undefined when 1 − tan²(θ) = 0, which happens at θ = 45° and θ = 135° (and their equivalents). The calculator will show “undefined” there.

Where these come up in practice: solving trig equations (rewriting 2θ terms so everything is in terms of θ), computing power-reduction formulas in integration, designing oscillator circuits where frequencies double, and in Fourier analysis where harmonics at twice the fundamental frequency appear naturally.

Deriving them takes one line each. Both fall straight out of the angle addition formulas by setting both angles equal. sin(A+B) = sinA cosB + cosA sinB, so with A = B = θ you get sin(2θ) = 2 sinθ cosθ. Same trick on cos(A+B) = cosA cosB − sinA sinB gives cos(2θ) = cos²θ − sin²θ. The other two cosine forms come from substituting sin²θ + cos²θ = 1 into that result, once for sin² and once for cos². Nothing to memorise beyond the addition formulas.

Reading them backwards is where they earn their keep. In integration you almost always use them in reverse: 1 − 2sin²θ = cos(2θ) rearranges to sin²θ = (1 − cos 2θ)/2, which turns an unintegrable square into something you can integrate term by term. Those are the power-reduction formulas, and they are the reason double angle identities appear in every calculus course rather than only in trigonometry.

A worked check at θ = 30°: sin 30° = 0.5 and cos 30° = 0.8660, so sin(2θ) = 2 × 0.5 × 0.8660 = 0.8660, which is sin 60°. And cos(2θ) = 0.8660² − 0.5² = 0.75 − 0.25 = 0.5, which is cos 60°. The calculator shows both routes so you can confirm it yourself.

The chart shows how sin(2θ) and cos(2θ) behave over a full cycle, with your own angle marked. Notice both complete two full oscillations between 0° and 360°, which is the “double angle” name doing exactly what it says.