Quick answer
Half-angle identities: sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2) and tan(θ/2) = sin θ/(1 + cos θ). For θ = 60°: sin 30° = 0.5, cos 30° = 0.8660 and tan 30° = 0.5774.
Enter an angle and its unit. The calculator returns sin, cos and tan of the half angle.
Related tools: Reference Angle Calculator: Acute angle between the terminal side and the x-axis. Coterminal Angle Calculator: Angles sharing the same terminal side. Angle of Elevation Calculator: Angle of elevation from a height difference and horizontal distance.
How to use the Half Angle Calculator
- Enter angle.
- Choose unit: Degrees and Radians.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Angle | A number | — |
| Unit | Choose one: Degrees; Radians | Starts at Degrees |
Formula and method
Formula
sin(θ/2) = ±√((1−cos θ)/2), cos(θ/2) = ±√((1+cos θ)/2)
IEEE-754 double precision; shown to at most 6 decimal places.
Evaluates sin, cos and tan at θ/2 directly (equivalent to the half-angle identities with the correct sign).
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Angle = 60; Unit = Degrees.
| Result | Value |
|---|---|
| sin(θ/2) | 0.5 |
| cos(θ/2) | 0.866025 |
| tan(θ/2) | 0.57735 |
The first line is the headline result (sin(θ/2)). Below it the half Angle Calculator lists cos(θ/2) and tan(θ/2), which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Choosing the sign
The ± depends on the quadrant of θ/2. For θ = 60° the half angle 30° is in the first quadrant, so all values are positive.
Where it is used
Half-angle formulas give exact values: cos 15° = √((1 + cos 30°)/2) = 0.9659. They also help in integration. tan is undefined where cos is zero.
How the result changes with angle
Holding every other input at the first worked example, this table changes only angle. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Angle | sin(θ/2) | cos(θ/2) | tan(θ/2) |
|---|---|---|---|
| 30 | 0.258819 | 0.965926 | 0.267949 |
| 60 | 0.5 | 0.866025 | 0.57735 |
| 120 | 0.866025 | 0.5 | 1.732051 |
| 240 | 0.866025 | -0.5 | -1.732051 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Angle: left empty shows "Enter angle." beside that field and no result.
Choosing between the Half Angle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
| [Coterminal Angle Calculator](/math/coterminal-angle/) | angle | Smallest non-negative coterminal angle |
| [Angle of Elevation Calculator](/math/angle-of-elevation-calculator/) | height difference and horizontal distance | Angle of elevation |
| [Angle of Depression Calculator](/math/angle-of-depression-calculator/) | height difference and horizontal distance | Angle of depression |
| [Double Angle Calculator](/math/double-angle-calculator/) | angle and unit | sin 2θ |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- tan undefined where cos is zero
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for angle to confirm this.
Frequently asked questions
What is cos 22.5°?
cos 22.5° = √((1 + cos 45°) ÷ 2) ≈ 0.9239.
How do I get the sign?
From the quadrant of the half angle.
Does it work in radians?
Yes. Choose radians as the unit and enter, for example, 1.0472 for 60°.
What does the Half Angle Calculator calculate?
Half-angle identities evaluated at an angle. It returns sin(θ/2), cos(θ/2) and tan(θ/2).
What formula does the Half Angle Calculator use?
sin(θ/2) = ±√((1−cos θ)/2), cos(θ/2) = ±√((1+cos θ)/2). Evaluates sin, cos and tan at θ/2 directly (equivalent to the half-angle identities with the correct sign).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. tan undefined where cos is zero.
What does the “Unit” option change?
It sets how the half Angle is worked out. The choices are Degrees and Radians.
What is an example of a half Angle result?
With angle 60 and unit Degrees, the sin(θ/2) is 0.5.
Related calculators
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Coterminal Angle Calculator — Angles sharing the same terminal side.
- Angle of Elevation Calculator — Angle of elevation from a height difference and horizontal distance.
- Angle of Depression Calculator — Angle of depression from a height difference and horizontal distance.
- Double Angle Calculator — Double-angle identities evaluated at an angle.
- Trigonometry Calculator — All six trigonometric functions of an angle.
- Half-Life Calculator — Remaining amount, elapsed time or half-life from exponential decay.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.