Formula and Method for the Half-Angle Identities
The half-angle identities let you find sin(θ/2), cos(θ/2), and tan(θ/2) from θ itself (or from cos θ, when θ is unknown but its cosine is). They come from the double-angle cosine identity written two ways: cos θ = 1 − 2sin²(θ/2) and cos θ = 2cos²(θ/2) − 1. Solving the first for sin²(θ/2) gives sin²(θ/2) = (1 − cos θ)/2, and solving the second for cos²(θ/2) gives cos²(θ/2) = (1 + cos θ)/2. Taking the square root of each produces the half-angle formulas: sin(θ/2) = ±√((1 − cosθ)/2) and cos(θ/2) = ±√((1 + cosθ)/2). Dividing the two — or substituting directly — gives the tangent form tan(θ/2) = sinθ/(1 + cosθ) = (1 − cosθ)/sinθ, which needs no separate sign check.
How the calculation works
Enter the angle θ and choose whether it is in degrees or radians. The calculator converts θ to radians, divides it by 2 to get the half angle, and evaluates sin(θ/2), cos(θ/2), and tan(θ/2) directly at that half angle — mathematically identical to applying the ± square-root identities above, but with the correct sign already built in because the exact half angle (and therefore its quadrant) is known. When cos(θ/2) is 0, i.e. θ/2 = 90° + k·180°, the tangent is undefined and the calculator reports that instead of a numeric value.
Resolving the ± sign by hand
If you only know cos θ (not θ itself) and must apply the square-root forms directly, the ± sign is fixed by locating which quadrant θ/2 falls in: sin(θ/2) is positive for θ/2 in quadrant I or II and negative in quadrant III or IV; cos(θ/2) is positive for θ/2 in quadrant I or IV and negative in quadrant II or III. Always determine the quadrant of θ/2 itself — not the quadrant of the original angle θ — since halving an angle can move it into a different quadrant.
Common mistakes
- Using the quadrant of θ instead of θ/2: a 300° angle sits in quadrant IV, but its half angle, 150°, sits in quadrant II — the signs for sin(θ/2) and cos(θ/2) must match 150°, not 300°.
- Dropping the ± sign entirely: the square root always returns a non-negative number; you must apply the correct sign yourself unless you use the sinθ/(1+cosθ) form for tangent.
- Mixing degrees and radians: make sure θ and the half-angle output are read in the same unit; 60° and 60 rad are very different angles.
Real-world applications
- Integral calculus uses the half-angle identities to rewrite sin²x and cos²x into forms that are easy to integrate.
- Optics and physics use half-angle relationships for phenomena like Snell's law derivations and phase calculations in wave interference.
- Surveying and navigation problems occasionally require bisecting a known angle to find intermediate bearings or sight lines.
- Engineering trigonometric simplifications rely on half-angle substitutions (the Weierstrass, or tangent half-angle, substitution) to convert trig equations into algebraic ones.