Half-Angle Calculator

Half-angle formulas let you determine trigonometric values for θ/2 from information about the full angle θ. They are a standard part of trigonometry because they bridge angle bisection, identities, and sign analysis in one compact set of formulas. The most common forms are sin(θ/2) = ±√((1 − cosθ)/2) and cos(θ/2) = ±√((1 + cosθ)/2). Once those are known, tan(θ/2) can be found from their ratio. This calculator automates that process and handles the sign choices for you. The sign is the subtle part. The square-root formulas only give magnitudes directly, because a square root is non-negative by definition. To decide whether sine or cosine should be positive or negative, you must know which quadrant the half-angle lies in. When the quadrant is set to Auto, the calculator determines the actual position of θ/2 from the entered angle after converting it to radians if necessary. That is usually the safest choice because it matches the geometric location implied by the input. When you choose a quadrant manually, you are selecting a specific sign branch of the identity, which is useful in textbook problems where extra information tells you where θ/2 must lie. Half-angle identities matter in both theoretical and applied work. They appear when simplifying expressions, proving other trigonometric identities, integrating powers of sine and cosine in calculus, and solving geometry problems involving angle bisectors. In analytic geometry and physics, they show up when converting between slope forms, modeling oscillations, or deriving formulas for rotation and projection. In computer graphics and signal processing, related trigonometric transformations often use the same algebraic relationships under the hood. One common mistake is forgetting that the input angle and the half-angle live in different quadrants. For example, θ may be in Quadrant III while θ/2 is in Quadrant II, so the signs of sine and cosine for the half-angle will not match the signs for θ itself. Another common mistake is mixing radians and degrees. A value of 90 means ninety degrees only if the unit selector is set to Degrees; otherwise it means ninety radians, which is a very different angle. Use this calculator whenever you need reliable half-angle values without manually tracking every sign case. It gives the numerical results immediately, keeps the unit choice explicit, and makes it easy to compare automatic sign detection with an imposed quadrant from a classroom exercise.