Half Angle Calculator - Halve Any Angle, Read the Three Identities

A half angle calculator is a trigonometry tool that halves any angle you enter and returns the three half-angle values sin(θ/2), cos(θ/2), and tan(θ/2). You type the original angle θ, choose degrees or radians, and the tool shows the halved angle in both units plus the three trig values that the half-angle identities describe.

• • Checking homework on half-angle identities: Students working through half-angle identity problems can plug the original angle in and confirm the three outputs they derived by hand, including the sign of each root.
• • Quick reference for sine, cosine, and tangent of half a known angle: Engineers, surveyors, and physics students who need sin, cos, and tan of a halved angle can read the three values from the result panel without re-deriving the identities from scratch.
• • Comparing the half-angle identity with the direct trig value: Readers learning the identities can see the calculator's output match the standard Math.sin, Math.cos, and Math.tan values of θ/2, which confirms that the identities are equivalent to the direct call.

Readers who already know the half-angle in degrees and need it in radians, gradians, or arcseconds can keep the same number in the Angle Converter and read every other unit side by side.

The tool converts your input to radians, divides by two, and applies the half-angle identities to write the trig values in terms of cos θ. The same identities are shown next to the formulas so you can read the rule that produced each output.

• θ (angle): Original angle. The calculator accepts it in degrees or radians and converts to radians before applying the identities.
• θ/2 (half angle): Halved angle. Shown in both degrees and radians so the reader can paste the value into another tool without re-converting.
• ± (sign of the root): Sign of the root is chosen by the quadrant of θ/2. Quadrant I and II give a positive sine, Quadrant III and IV give a negative sine; cosine is positive in Quadrant I and IV, negative in II and III.

When θ sits inside a textbook problem the identities turn the half-angle into a problem in cos θ alone, which means a single cosine value drives all three outputs. The calculator first halves the angle, then computes sin(θ/2), cos(θ/2), and tan(θ/2) using the identities, and finally rounds the result panel values for display.

According to Weisstein, Eric W. "Half-Angle Formulas," MathWorld, the half-angle identities rewrite sin(θ/2) and cos(θ/2) as square roots of expressions in cos θ, with the sign chosen by the quadrant of θ/2.

Readers who start with a radian answer and want the same number expressed in degrees for a homework step can pipe the half-angle radian value into the Radians to Degrees Calculator for a single-number conversion.

Four short definitions keep the formulas honest. None of them depend on degrees versus radians, so the same identities hold whether the reader typed 60 or π/3.

Readers who want to go the other way and recover θ from a half-angle tangent value can pair the result with the Arctan Calculator, which inverts tangent into the principal angle.

Type the original angle, pick its unit, and read the half angle and the three trig values. The result panel updates as you type, so there is no submit step to remember.

1. 1 Enter the original angle: Type the angle θ in the Angle θ box. Positive values rotate counter-clockwise, negative values rotate clockwise.
2. 2 Pick degrees or radians: Choose the unit you typed. If your source uses π form like π/3, switch to Radians and type 1.0471976 for the radian value.
3. 3 Read the half angle: The first two result rows show θ/2 in degrees and radians. Copy whichever unit matches the next tool or your homework step.
4. 4 Read the three half-angle values: The next three rows give sin(θ/2), cos(θ/2), and tan(θ/2). The sign of each value is the principal value, which is what the identity returns for the quadrant of θ/2.
5. 5 Cross-check with the direct call: If a downstream step uses Math.sin(θ/2) directly, the result panel value matches it to the six-decimal display precision.

A reader needs sin, cos, and tan of 15° for a geometry proof. Because the tool halves whatever angle is typed in, they enter 30 into the Angle θ box, leave the unit on Degrees, and read sin(15°) ≈ 0.258819, cos(15°) ≈ 0.9659258, tan(15°) ≈ 0.2679492. The Half Angle in Degrees row confirms 15° and the Half Angle in Radians row shows 0.2617994 rad, so the reader can paste either unit into the next step without re-converting.

Readers who need to recover θ from cos(θ/2) can pass the cosine output of this tool into the Arccos Calculator, which converts the cosine back to the principal angle.

A short list of what the tool does well, and what it is not designed to do, helps you put the result in the right place in your day.

• • Three trig values in one step: The calculator returns sin(θ/2), cos(θ/2), and tan(θ/2) at the same time, so a single read covers the three half-angle identity outputs.
• • Degrees and radians in the same panel: The half angle is shown in both units, so the reader can paste the value into a trigonometry, surveying, or physics problem without re-converting.
• • Sign chosen by the quadrant of θ/2: The square-root form of the identities needs a sign, and the calculator applies the quadrant rule for θ/2 to pick it. Readers do not have to remember the sign by hand.
• • Worked example included: A short 60° worked example sits next to the formulas, so the reader can see which output came from which identity without doing the algebra.

Readers who want the inverse half-angle identity can take the sine output of this tool and pass it into the Arcsin Calculator, which returns the principal angle whose sine equals the value.

The tool is honest about which factors change the number, which factors only change the unit, and which factors it cannot see at all.

• • The tool returns principal values for sin(θ/2) and cos(θ/2). Problems that ask for the negative root must add the sign by hand using the quadrant of θ/2.
• • Half-angle identities describe real-valued half angles. The Undefined row for tan(θ/2) is a signal that the identity has no real answer at that input, not a JavaScript artifact; treat the result as a vertical asymptote in any downstream graph or table.

According to OpenStax Algebra and Trigonometry 2e, Section 9.3, the half-angle identities follow from the double-angle identity cos(2α) = 1 - 2 sin²α, solved for sin α and cos α.

Readers who need the half-angle in a surveyor's format can pipe the degree output into the Degrees Minutes Seconds Calculator, which splits the decimal degree value into degrees, minutes, and seconds for mapping, navigation, or written work.