Free · Algebra and geometric models

Special-Angle Ratios and Numerical Values

Calculate six trigonometric ratios with exact special-angle expressions and undefined-pole notes.

  • Formula & worked example
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Calculator inputs

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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.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

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

sin θ, cos θ, tan θ and reciprocal ratios

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

  • Angles are periodic. Inverse functions use their principal branches.
  • Tangent/secant are undefined at odd quarter turns; cotangent/cosecant at half turns.

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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