Inverse Cosine CalculatorFree online math calculator — instant results, no sign-up
Calculate the principal inverse cosine in degrees.
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About the Inverse Cosine Calculator
The Inverse Cosine Calculator finds the principal angle in degrees from a given cosine value. Enter a value from −1 to 1 to obtain an angle from 0° to 180°. With the default cosine value of 0.5, the calculator returns 60°.
- Cosine value
- The dimensionless cosine of the angle you want to find, which must be between −1 and 1 inclusive.
Formula and method
θ = arccos(x) × 180/π; for x = 0.5, θ = (π/3) × 180/π = 60°.
- xThe entered cosine value, a dimensionless number in [−1, 1].
- θThe principal inverse cosine angle in degrees, in [0°, 180°].
- arccos(x)The principal angle in radians whose cosine is x.
- πThe mathematical constant pi; π radians equals 180°.
Check that the input is a real number in [−1, 1]. Evaluate its principal inverse cosine in radians, then multiply by 180/π to convert to degrees. For the default input 0.5, arccos(0.5) is π/3 radians, giving 60°.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Cosine value0.5
Results
- Result60
How to use the Inverse Cosine Calculator
- 1Identify the cosine value of the angle you want to recover.
- 2Enter it in the Cosine value field, or leave the default value of 0.5.
- 3Confirm that the value lies between −1 and 1 inclusive.
- 4Run the calculation to obtain the principal inverse cosine in degrees.
- 5Read the Result row: with 0.5 entered, Result = 60 means an angle of 60°.
Reading your result
The Result row shows the principal angle in degrees whose cosine equals the input. For the default input of 0.5, Result = 60 means cos(60°) = 0.5. This is the unique matching angle within 0° to 180°, not a list of all angles with that cosine.
Limitations
Real-valued inverse cosine is defined only for inputs from −1 to 1 inclusive; values outside that interval have no real angle result. The output is restricted to the principal range of 0° to 180° and does not recover an original angle outside that range. Rounded inputs and finite numerical precision can affect the result, especially near −1 and 1.
Common questions
Why does the default cosine value of 0.5 return 60?+
Because cos(60°) = 0.5, and 60° lies within the principal inverse cosine range of 0° to 180°. The displayed result is in degrees.
Does cos⁻¹ mean inverse cosine or the reciprocal of cosine?+
In this calculator, cos⁻¹ means arccos: finding an angle from its cosine. The reciprocal, 1/cos(θ), is a different function called secant.
Can I enter a negative cosine value?+
Yes, provided it is no less than −1. A negative cosine value produces a principal angle greater than 90° and no greater than 180°.
Why does the calculator return only one angle when cosine repeats?+
Inverse cosine uses a restricted range of 0° to 180° so each valid input has exactly one principal result. Other matching angles can be expressed as ±θ + 360°k, where θ is the result and k is any integer.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.