Half-Angle Calculator
Welcome to Calcugo's half-angle calculator, where we study half-angle trigonometric identities. While double-angle formulas let you find functions of twice an angle, half-angle identities go the other way — they find sin, cos, and tan of half the angle using the power reducing identities.
Trigonometric Functions
Trigonometry relates a triangle's side lengths to its angles. For a right triangle with angle α, opposite side a, adjacent side b, and hypotenuse c:
- sin α = a / c (opposite / hypotenuse)
- cos α = b / c (adjacent / hypotenuse)
- tan α = a / b (opposite / adjacent)
These definitions extend beyond 0°–90° by placing a point A = (x, y) on the coordinate plane and measuring the directed angle α counterclockwise from the positive x-axis. This makes negative angles and angles greater than 360° perfectly valid.
Half-Angle Trig Identities
Starting from the power reducing formulas:
- sin²(x) = (1 − cos 2x) / 2
- cos²(x) = (1 + cos 2x) / 2
- tan²(x) = (1 − cos 2x) / (1 + cos 2x)
Replacing x with x/2 (so 2x becomes x) and taking square roots gives the half-angle identities:
- sin(x/2) = ±√((1 − cos x) / 2)
- cos(x/2) = ±√((1 + cos x) / 2)
- tan(x/2) = ±√((1 − cos x) / (1 + cos x)) = sin x / (1 + cos x) = (1 − cos x) / sin x
The ± sign depends on the quadrant of x/2:
- Quadrant I (x/2 in 0°–90°): sin +, cos +, tan +
- Quadrant II (x/2 in 90°–180°): sin +, cos −, tan −
- Quadrant III (x/2 in 180°–270°): sin −, cos −, tan +
- Quadrant IV (x/2 in 270°–360°): sin −, cos +, tan −
This calculator uses the direct formula tan(x/2) = sin x / (1 + cos x), which automatically carries the correct sign without needing to determine the quadrant separately.
How to Use This Calculator
- Select the angle unit: degrees (°) for the American/common system, radians (rad) for the metric/scientific system.
- Enter the full angle x.
- Click Calculate — the calculator instantly returns sin(x/2), cos(x/2), and tan(x/2) along with the intermediate value cos(x) and the full formulas.
Example: Half-Angle of 60°
Let x = 60°. Then x/2 = 30°.
- cos 60° = 0.5
- sin(30°) = √((1 − 0.5)/2) = √(0.25) = 0.5 ✓
- cos(30°) = √((1 + 0.5)/2) = √(0.75) ≈ 0.866025 ✓
- tan(30°) = sin 60° / (1 + cos 60°) = (√3/2) / (3/2) = 1/√3 ≈ 0.577350 ✓
Angle Units
Degrees divide a full circle into 360 equal parts — familiar from everyday geometry, navigation, and engineering (American/Imperial context).
Radians measure angles by arc length on the unit circle. One full circle = 2π rad. Radians are the standard unit in higher mathematics, physics, and the metric system.
Conversion: 1 rad = 180°/π ≈ 57.2958° | 1° = π/180 rad ≈ 0.017453 rad
Common Half-Angle Values
| x | x/2 | sin(x/2) | cos(x/2) | tan(x/2) |
|---|---|---|---|---|
| 0° | 0° | 0 | 1 | 0 |
| 30° | 15° | ≈ 0.2588 | ≈ 0.9659 | ≈ 0.2679 |
| 45° | 22.5° | ≈ 0.3827 | ≈ 0.9239 | ≈ 0.4142 |
| 60° | 30° | 0.5 | ≈ 0.8660 | ≈ 0.5774 |
| 90° | 45° | ≈ 0.7071 | ≈ 0.7071 | 1 |
| 120° | 60° | ≈ 0.8660 | 0.5 | ≈ 1.7321 |
| 180° | 90° | 1 | 0 | undefined |
| 270° | 135° | ≈ 0.7071 | ≈ −0.7071 | −1 |
| 360° | 180° | 0 | −1 | 0 |
FAQs
What are half-angle identities used for?
They appear in integral calculus (Weierstrass substitution), Fourier analysis, optics, and signal processing — wherever you need to split an angle in half algebraically.
Why is there a ± sign in the formulas?
Because taking a square root can yield a positive or negative result. The correct sign is determined by the quadrant that x/2 falls in. Our calculator resolves this automatically.
What happens when tan(x/2) is undefined?
tan(x/2) is undefined when cos(x) = −1, i.e., when x = 180° + 360°k. This corresponds to x/2 = 90° + 180°k, where the tangent function is undefined.
Can I enter negative angles?
Yes. Negative angles represent clockwise rotation. The calculator handles any real value.