How this calculator works
The half-angle square roots give magnitudes; the signs depend on the quadrant of the half-angle. Supplying the full signed angle resolves that choice directly. When only a function value is known, choose the quadrant of the angle at which that value was measured. The tool reconstructs an angle in the corresponding turn and rejects inconsistent signs. Axis endpoints can be selected from either neighbouring quadrant.
Formula
sin(x/2)=±√[(1-cos x)/2]; cos(x/2)=±√[(1+cos x)/2]; tan(x/2)=sin x/(1+cos x) where defined. Signs follow x/2.
Worked example
At x = 60°, sin 30° = 0.5 and cos 30° = √3/2. At x = 300°, the half-angle is 150°, so cosine is negative even though cos 300° is positive.
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Frequently asked questions
Why do the formulas show ±?
Taking a square root gives a non-negative magnitude. The position of the half-angle selects its sign.
Does knowing cos x identify x/2 uniquely?
No. A quadrant and a revolution convention are needed. Function-value modes use a known angle within one turn.
Can I enter a full angle above 360°?
Yes in full-angle mode. The calculator halves the actual signed angle, so the revolution count can change the answer.