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Half Angle Formulas Calculator

Online half-angle calculator: enter α to find sin(α/2) and cos(α/2), with the sign set by the half-angle quadrant.

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sin(α/2)=√((1−cosα)/2)
Positive root used

📖 Tutorial | Half Angle

1. Definition
Opening the power-reduction formulas: sin(α/2)=±√((1−cosα)/2), cos(α/2)=±√((1+cosα)/2).
2. Formula & Notation
αOriginal angle
α/2Half angle
3. Properties & Laws
  • sin(α/2)=±√((1−cosα)/2);
  • cos(α/2)=±√((1+cosα)/2);
  • tan(α/2)=sinα/(1+cosα);
  • Sign from the half-angle quadrant.
4. Steps
  1. Enter α (degrees);
  2. Click "Calculate" (positive root);
  3. Or click "Load Example".
5. Worked Examples
Example: α=60° gives sin30=√(0.5/2)=0.5.
6. Common Pitfalls
The ± sign follows the α/2 quadrant;
Sine uses 1−cos, cosine 1+cos;
Inverse to the double-angle formulas.

❓ FAQ | Half Angle

Sine?
√((1−cosα)/2).
Cosine?
√((1+cosα)/2).
Sign?
By quadrant.
Tangent?
sinα/(1+cosα).
Plus or minus inside?
Sine subtracts.
Double angle?
Inverse.