Free · Algebra and geometric models

Six Circular Functions from an Angle

Explore angle ratios, reciprocal functions and special-angle radical forms.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For the worked inputs, Tan print = 1, Cot print = 1.

sin θ=opposite/hypotenuse; cos θ=adjacent/hypotenuse; tan θ=sin θ/cos θ

Worked example

Enter these known values and leave the other values blank.

Angle
45 deg
Result
At 45 degrees modulo a full turn, the exact ratios are sin=cos=√2/2, tan=cot=1, sec=csc=√2. Angle degrees print = 44.99999999999999999999999999999999999999999999999999999999999999. Sin print = 0.7071067811865475244008443621048490392848359376884740365883398689. Angle degrees print = 44.99999999999999999999999999999999999999999999999999999999999999. Cos print = 0.707106781186547524400844362104849039284835937688474036588339869. Angle degrees print = 44.99999999999999999999999999999999999999999999999999999999999999. Tan print = 0.9999999999999999999999999999999999999999999999999999999999999998. Angle degrees print = 44.99999999999999999999999999999999999999999999999999999999999999. Cot print = 1. Angle degrees print = 44.99999999999999999999999999999999999999999999999999999999999999. Sec print = 1.414213562373095048801688724209698078569671875376948073176679738. Angle degrees print = 44.99999999999999999999999999999999999999999999999999999999999999. Csc print = 1.414213562373095048801688724209698078569671875376948073176679738.

Assumptions and limitations

  • For arbitrary angles, signed circular functions follow unit-circle quadrants.
  • Reciprocal functions are undefined when their denominator is zero. Principal inverse branches are retained.

Common questions

What must I enter?

Choose the mode when available and enter its independent known quantities. Leave an unknown editable quantity empty to use a supported inverse.

How should I use the result?

Interpret the numbers within the stated mathematical assumptions. Available unit selectors convert quantities; display rounding does not replace the internal calculation.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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