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Induction Formulas Calculator

Online reduction formula calculator: enter any angle to get the reference acute angle and sin, cos, reducing any angle to an acute-angle function.

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Reduce to an acute angle; sign by quadrant
全+sin+tan+cos+
May be negative or over 360

📖 Tutorial | Reduction

1. Definition
Using periodicity and symmetry, reduce any angle to an acute-angle function. The name changes for odd multiples of 90°, and the sign follows the original quadrant.
2. Formula & Notation
k·90°±αDecomposition
Reference angleSmallest angle to the x-axis
3. Properties & Laws
  • For odd k, sin↔cos swap; even k leaves the name;
  • Take the sign from the original quadrant;
  • sin(180°+α)=−sin α;
  • cos(−α)=cos α;
  • Periods: sin/cos 360°, tan 180°.
4. Steps
  1. Enter an angle (degrees, may be negative);
  2. Click "Calculate" for the reference angle and sin, cos;
  3. Or click "Load Example".
5. Worked Examples
Example: 210°=180°+30° gives sin=−0.5, cos≈−0.8660.
6. Common Pitfalls
Fix the name first, then the sign;
The reference angle is measured to the x-axis;
Normalize negative angles first.

❓ FAQ | Reduction

Rule?
Name by the multiple of 90°, sign by quadrant.
Reference angle?
Smallest angle to the x-axis.
sin(180+α)?
−sin α.
cos(−α)?
cos α.
Quadrant signs?
ASTC: all, sine, tangent, cosine.
Period?
360° for sin/cos.
Why reduce?
To evaluate via tables or a calculator.