Free · Geometry and function relationships

Arctangent Principal Angle

Arctangent converts a real tangent value to its principal angle strictly between −90° and 90°. A tangent ratio does not identify the angle’s quadrant without other information.

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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 x = 1, y = 45.

θ=atan(x)

Worked example

Enter these known values and leave the other values blank.

x
1
y
45 deg

Assumptions and limitations

  • Arctangent converts a real tangent value to its principal angle strictly between −90° and 90°. A tangent ratio does not identify the angle’s quadrant without other information.
  • Clear the unknown field before solving. Locks preserve known quantities; an inverse may have multiple solutions and this model selects its principal branch.
  • Numerical rounding does not describe measurement uncertainty. Extremely large arguments or near-singular inverses can exceed the available precision; use moderate values for interpretable finite results.

Common questions

How do I enter a consistent set of values?

Arctangent converts a real tangent value to its principal angle strictly between −90° and 90°. A tangent ratio does not identify the angle’s quadrant without other information.

Can I calculate a missing input?

Clear that field and provide enough independent known quantities. Output-only fields and nonunique inverse branches may need a different mode.

References

The principal arctangent angle is strictly between −π/2 and π/2. The source rounds these bounds to ±1.5708, incorrectly accepting endpoints and a small outside interval.

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