Half Angle Calculator

Compute half‑angle values for any angle. Automatic sign determination based on the quadrant of θ/2. Includes formulas, special angles and step‑by‑step explanations.

Core formulas (θ any angle):

sin(θ/2) = ±√((1−cosθ)/2)   cos(θ/2) = ±√((1+cosθ)/2)
tan(θ/2) = ±√((1−cosθ)/(1+cosθ)) = (1−cosθ)/sinθ = sinθ/(1+cosθ)

30° 45° 60° 90° 120° 180° π rad π/4

Understanding Half‑Angle Formulas

Half‑angle formulas express trigonometric functions of θ/2 in terms of the cosine of the original angle θ. They are essential in calculus (integration), solving trigonometric equations, physics (wave phenomena), and geometry.

Core formula set:

sin²(θ/2) = (1−cosθ)/2 → sin(θ/2) = ±√((1−cosθ)/2)

cos²(θ/2) = (1+cosθ)/2 → cos(θ/2) = ±√((1+cosθ)/2)

tan²(θ/2) = (1−cosθ)/(1+cosθ) → tan(θ/2) = ±√((1−cosθ)/(1+cosθ))

Additionally, tan(θ/2) has two root‑free forms:
tan(θ/2) = (1−cosθ)/sinθ = sinθ/(1+cosθ)

1. Derivation (from double‑angle identities)

Start from the double‑angle formulas for cosine: cos(2α) = 1 − 2sin²α and cos(2α) = 2cos²α − 1. Set α = θ/2, then cosθ = 1 − 2sin²(θ/2) → sin²(θ/2) = (1−cosθ)/2, and cosθ = 2cos²(θ/2) − 1 → cos²(θ/2) = (1+cosθ)/2. Dividing these gives the formula for tan²(θ/2).

2. Sign determination (quadrant of θ/2)

The “±” sign depends on where θ/2 lies. First normalize θ to the range 0°–360° (0–2π). Then θ/2 will be between 0° and 180°. Use the sign rules below:

  • Quadrant I (0°–90°): sin +, cos +, tan +
  • Quadrant II (90°–180°): sin +, cos –, tan –
  • Quadrant III (180°–270°): sin –, cos –, tan +
  • Quadrant IV (270°–360°): sin –, cos +, tan –

When θ/2 lies exactly on an axis (0°, 90°, 180°), the signs follow the usual definitions (e.g., sin0°=0, tan90° undefined).

3. Exact values for special angles

The table below lists common angles θ and their half‑angle exact expressions (where possible):

θ (°) cosθ sin(θ/2) cos(θ/2) tan(θ/2)
0° 1 0 1 0
30° √3/2 ≈0.8660 √((1−√3/2)/2) = √((2−√3)/4) ≈0.2588 √((1+√3/2)/2) = √((2+√3)/4) ≈0.9659 2−√3 ≈0.2679
45° √2/2≈0.7071 √((1−√2/2)/2) = √((2−√2)/4) ≈0.3827 √((1+√2/2)/2) = √((2+√2)/4) ≈0.9239 √2−1 ≈0.4142
60° 0.5 0.5 √3/2≈0.8660 1/√3≈0.5774
90° 0 √2/2≈0.7071 √2/2≈0.7071 1
120° −0.5 √3/2≈0.8660 0.5 √3≈1.7321
135° −√2/2≈−0.7071 √((1+0.7071)/2)≈0.9239 √((1−0.7071)/2)≈0.3827 √2+1≈2.4142
150° −√3/2≈−0.8660 0.9659 0.2588 2+√3≈3.7321
180° −1 1 0 undefined (∞)

(cot, sec, csc can be obtained as reciprocals.)

4. Applications

  • Integration: The substitution t = tan(θ/2) (Weierstrass substitution) converts rational trigonometric integrals into rational functions.
  • Physics: Wave interference, Fresnel equations, and optics frequently involve half‑angles.
  • Geometry: Chord lengths and inscribed angles in circles are simplified using half‑angle relations.
  • Signal processing: Modulation and demodulation often use trigonometric half‑angle identities.

5. Historical note

Half‑angle formulas were used by Ptolemy to construct chord tables. Indian mathematicians like Aryabhata (5th century) and Bhaskara (12th century) developed trigonometry further, and half‑angle relations were fundamental. Today they are a standard part of high school and college mathematics.

Using this calculator:

  • Any real number (positive or negative) is accepted; the angle is automatically normalized to 0°–360°.
  • For radian input, you can use expressions like pi/4 (math.js is loaded).
  • The sign is based solely on the quadrant of θ/2, not θ.
  • If a function is undefined (e.g., tan 90°), the calculator displays “undefined”.

Frequently Asked Questions

The function values are taken from the standard definitions (e.g., sin0°=0, cos90°=0, tan90° undefined). The sign is conventional.

We first normalize θ to the range 0°–360° by adding/subtracting multiples of 360° (or 2π). Then θ/2 is between 0° and 180°, and its quadrant is determined from that normalized half‑angle.

They are algebraically equivalent. The root‑free forms are often more convenient because they avoid sign ambiguity and are easier to differentiate or integrate.

It returns high‑precision decimal approximations. For exact symbolic values (e.g., √3/2) please refer to the special‑angle table above.