Half Angle Calculator

This half angle calculator works out sin(x/2), cos(x/2) and tan(x/2) for any angle x, using the standard half angle identities from trigonometry. Enter your angle, select whether it is in degrees or radians, and the calculator instantly returns all three half-angle values, with sin(x/2) shown as the main result and cos(x/2) and tan(x/2) alongside it. The textbook half angle formulas write these functions as square roots of expressions involving cos x, and those square roots carry a plus or minus sign that depends on which quadrant the half angle falls in, which is a common source of errors when working by hand. This tool sidesteps that problem entirely by converting your angle to the actual half angle in radians and evaluating each function directly, so the sign is always correct without you needing to reason about quadrants. It is useful for trigonometry and calculus students checking exact values, working through the tangent half angle substitution used in integration, or simply verifying homework and exam answers quickly. Try it with 60 degrees to see the worked example reproduced: sin(30) = 0.5, cos(30) = 0.866025 and tan(30) = 0.577350. Results are rounded to six decimal places for display, and tan(x/2) is reported as undefined where cos(x/2) is zero.

Angle

Angle unit

sin(x/2) = 0.5

cos(x/2)0.866025

tan(x/2)0.57735

The formula

The half angle identities are: sin(x/2) = ± sqrt((1 − cos x) / 2); cos(x/2) = ± sqrt((1 + cos x) / 2); and tan(x/2) = (1 − cos x) / sin x = sin x / (1 + cos x). The sign depends on the quadrant of x/2.

Worked example

For x = 60 degrees: sin(x/2) = sin 30 = 0.5, cos(x/2) = cos 30 = 0.866025, tan(x/2) = tan 30 = 0.577350. Enter 60 degrees to confirm.

Frequently asked questions

What is the half angle formula for sine?

sin(x/2) = plus or minus the square root of (1 minus cos x) over 2, with the sign set by the quadrant of x over two.

How do I avoid the sign ambiguity?

Work out which quadrant x over two is in, or, as this calculator does, evaluate the function of the actual half angle directly.

What is the tangent half angle form?

tan(x/2) = (1 minus cos x) divided by sin x, which also equals sin x divided by (1 plus cos x).

Who this calculator is for

This calculator is for trigonometry and calculus students using the half angle identities.

What this calculator assumes

You enter one angle in the chosen unit.
The half angle is evaluated in radians.
Results are rounded for display.

Formula and sources

Related calculators

Trig Calculator: all six trig functions.
Sine Calculator: sine of an angle.
Right Angle Triangle Calculator: solve a right triangle.

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