Half Angle Formula Calculator

Work out half angle formula instantly with clear inputs, formula shown and shareable results.

About Half Angle Formula Calculator

Rearranging cos 2x = 1 − 2sin²x gives the half-angle formulas sin(θ/2) = ±√((1 − cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2), with the sign fixed by the quadrant of θ/2.

Formulas

Sine

sin(θ/2) = ±√((1 − cos θ) / 2)

Cosine

cos(θ/2) = ±√((1 + cos θ) / 2)

Tangent

tan(θ/2) = sin θ / (1 + cos θ)

Worked example

With the default values below, Half Angle Formula Calculator returns sin(θ/2) of 0.50000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Angle θ: 60
Angle unitone of Degrees, Radians, Gradians: Degrees

Results

sin(θ/2): 0.50000000
cos(θ/2): 0.86602540
tan(θ/2) (0 at the asymptote): 0.57735027
sin(θ/2) via √((1 − cos θ)/2): 0.50000000

Step by step

1Values used
2Sine
3Cosine
4Tangent
5sin(θ/2)= 0.50000000
6cos(θ/2)= 0.86602540
7tan(θ/2) (0 at the asymptote)= 0.57735027
8sin(θ/2) via √((1 − cos θ)/2)= 0.50000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For θ = 60°?

sin 30° = 0.5 and cos 30° = 0.8660254, and the radical form √((1 − 0.5)/2) = 0.5 agrees.

How do I choose the sign?

Look at which quadrant θ/2 lands in; the radical only gives the magnitude.