Half-Angle Calculator

Half-angle calculator work starts with a deceptively simple question: what do the trigonometric ratios become after an angle is cut in two? That question appears in exact trig tables, integration substitutions, phasor analysis, and geometry proofs. This page explains the identities used by the calculator, the sign convention that prevents wrong-quadrant answers, and the rounding you see in the displayed results.

How the calculator follows the identity

The calculator accepts one angle in degrees, limited to the inclusive range from -360 to 360. It first converts the entered angle to radians because the underlying sine, cosine, and tangent functions expect radians. It then evaluates the original angle’s sine, cosine, and tangent. For tangent of the original angle, the method checks whether the original cosine is extremely close to zero; if so, it reports tangent as not defined.

The half-angle values are then computed from the cosine of the original angle. This mirrors the standard identities derived from the double-angle formulas. The subtle part is the sign. A square root cannot tell whether the final sine or cosine should be positive or negative. The calculator halves the degree measure, reduces that half angle into a 0 to 360 degree cycle, assigns a quadrant, and applies the usual quadrant signs: sine positive in quadrants I and II, cosine positive in I and IV, and tangent positive in I and III.

That means a negative input is not treated as an afterthought. An input of -120 degrees produces a half angle of -60 degrees, which reduces to quadrant IV; sine is negative, cosine is positive, and tangent is negative. Always decide the sign after halving.

Formula and variables

The double-angle identity for cosine gives the half-angle identities. Starting with

$\cos x = 1 - 2\sin^2\left(\frac{x}{2}\right)$

solve for the squared sine of the half angle:

$\sin^2\left(\frac{x}{2}\right) = \frac{1 - \cos x}{2}$

Taking the square root gives the signed sine formula:

$\sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos x}{2}}$

The companion cosine identity comes from

$\cos x = 2\cos^2\left(\frac{x}{2}\right) - 1$

so

$\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos x}{2}}$

For tangent, divide the signed sine half-angle value by the signed cosine half-angle value. The magnitude form used by the calculator is

$\tan\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos x}{1 + \cos x}}$

Here, $x$ is the original input angle in degrees as shown to the user; the trigonometric functions are evaluated after converting that angle to radians. The symbols $\sin$, $\cos$, and $\tan$ are dimensionless ratios, so their outputs have no units. Degrees measure angle size, not length.

Worked example: 120 degrees

Suppose the input angle is 120 degrees. The calculator halves it first, so the result labels use 60 degrees. The original cosine is

$\cos(120^\circ) = -0.5$

For sine of the half angle, substitute into the identity:

$\sin\left(\frac{120^\circ}{2}\right) = \sqrt{\frac{1 - (-0.5)}{2}} = \sqrt{0.75} = 0.866025\ldots$

The half angle is 60 degrees, quadrant I, so the positive sign is used. For cosine,

$\cos\left(60^\circ\right) = \sqrt{\frac{1 + (-0.5)}{2}} = \sqrt{0.25} = 0.5$

For tangent,

$\tan(60^\circ) = \sqrt{\frac{1 - (-0.5)}{1 + (-0.5)}} = \sqrt{3} = 1.732050\ldots$

The displayed outputs therefore round to sin 60 degrees = 0.866, cos 60 degrees = 0.5, and tan 60 degrees = 1.7321. The original-angle items show sin 120 degrees = 0.866, cos 120 degrees = -0.5, and tan 120 degrees = -1.7321. Those signs differ because tangent at 120 degrees is in quadrant II, while the half angle is in quadrant I.

Interpretation and applications

Half-angle identities reduce a trigonometric expression involving a smaller angle to one involving a known larger angle. In calculus, they are part of power-reduction formulas that turn squared sine or cosine terms into expressions easier to integrate. In geometry, they support exact constructions such as deriving the values at 15 degrees from 30 degrees or at 22.5 degrees from 45 degrees. In signal work, they help compare phase shifts when an angle is split between two components.

For unit conversion before using the calculator, visit the radians to degrees calculator or the broader angle converter. For geometry settings where a trigonometric angle comes from a line, the slope calculator can help connect rise-over-run with tangent. Very small or very large computed values are easier to report with the scientific notation calculator.

Edge cases and common mistakes

The calculator rejects invalid numeric values and any angle less than -360 or greater than 360. At 180 degrees, the half angle is 90 degrees: sine is 1, cosine is 0, and tangent is not defined. The tangent half-angle formula also has a denominator of $1 + \cos x$, so an input where $\cos x = -1$ needs special care. The calculator catches this by returning tangent as not defined.

Do not use the quadrant of the original angle to select the sign. A 240 degree input has a 120 degree half angle, so sine is positive and cosine is negative even though the original angle lies in quadrant III. Also avoid rounding the original cosine before the square roots; small changes near 1 or -1 can decide whether a numerator or denominator is essentially zero. Finally, remember that the displayed values are decimal approximations, not exact symbolic simplifications.

Sources

• OpenStax, Precalculus 2e: Double-Angle, Half-Angle, and Reduction Formulas — derivation and use of half-angle identities.
• NIST Digital Library of Mathematical Functions, Trigonometric Functions — reference definitions and identities for trigonometric functions.