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Inverse Cosine with Principal-Angle References

Inverse Cosine with Principal-Angle References. Find the principal inverse angle or convert a entered angle to its sine/cosine ratio.

  • 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

The principal inverse cosine of 0.5 is 60°, equal to π/3 radians.

y = arccos(x)

Worked example

Enter these known values and leave the other values blank.

Cosine ratio x
.5
Angle y
60 deg

Assumptions and limitations

  • Arithmetic uses 64 significant decimal digits, with separate rounding for displayed results. Very large integer arithmetic is not an exact arbitrary-length integer service.
  • The ratio x must lie from −1 through 1. The inverse returns the principal arccosine angle from 0° through 180°.
  • The angle selector converts between the available units. Entering an angle evaluates its forward trigonometric ratio; angles outside the principal range do not have the same inverse angle.

Common questions

Is inverse cosine or sine the reciprocal of cosine or sine?

No. An inverse trigonometric function finds an angle; a reciprocal divides one by a trigonometric ratio.

Can I enter the result to calculate a missing input?

The retained inverse rules solve from sufficient known inputs. Roots and inverse trigonometric functions use principal values; indeterminate inputs do not have a unique solution.

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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