Last updated: July 31, 2026
Inverse Cosine Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
The inverse cosine returns the angle whose cosine is x. Its real-number domain is [-1, 1] and its principal range is [0°, 180°] or [0, π] radians.
Where:
- x=Input cosine value
- \\theta=Inverse cosine angle in radians(radians)
- \\theta_{deg}=Inverse cosine angle in degrees(degrees)
Worked Examples
Maximum cosine value
cos(0) = 1, so the inverse cosine of 1 is 0.
- 1Check that x = 1 is inside the domain.
- 2Compute arccos(1) = 0 radians.
- 3Convert 0 radians to 0 degrees.
Right-angle cosine
A cosine value of 0 corresponds to a 90-degree principal angle.
- 1Check that x = 0 is in the domain.
- 2Compute arccos(0) = π/2 ≈ 1.570796 radians.
- 3Convert π/2 to 90 degrees.
Minimum cosine value
cos(π) = -1, so the inverse cosine of -1 is π radians.
- 1Check that x = -1 is inside the domain.
- 2Compute arccos(-1) = π ≈ 3.141593 radians.
- 3Convert π radians to 180 degrees.
Special-angle cosine
The cosine of 60° is 0.5, so arccos(0.5) returns 60°.
- 1Check that x = 0.5 is inside the domain.
- 2Compute arccos(0.5) = π/3 ≈ 1.047198 radians.
- 3Convert π/3 to 60 degrees.
Introduction
The inverse cosine function, written arccos or cos⁻¹, answers the question: “Which angle has cosine equal to x?” This calculator returns the principal angle in both radians and degrees, provided the input lies within the real-number cosine domain from -1 to 1.
What inverse cosine means
Inverse cosine reverses the cosine function on its principal interval. Instead of starting with an angle and finding a ratio, you start with the cosine value and recover the matching principal angle.
- Input:
a cosine value x.
- Output:
the angle θ such that cos(θ) = x.
- Principal output range:
0 to π radians.
Domain and range
A real cosine value can never be smaller than -1 or larger than 1, so those are the valid input limits. The standard inverse cosine output is restricted to angles from 0° to 180°.
- Domain:
-1 ≤ x ≤ 1.
- Range in radians:
0 ≤ θ ≤ π.
- Range in degrees:
0° ≤ θ ≤ 180°.
Values outside [-1, 1] have no real inverse cosine result.
Converting radians to degrees
Most programming languages compute inverse trigonometric functions in radians. To make the answer easier to interpret, the calculator also converts that radian result into degrees.
Find radians with arccos(x).
Multiply by 180/π.
Round the displayed values for readability.
Special inverse cosine values
Several inputs produce familiar benchmark angles that are useful in trigonometry and precalculus.
arccos(1) = 0°
arccos(0) = 90°
arccos(0.5) = 60°
arccos(-1) = 180°
Where inverse cosine is used
Inverse cosine appears in right-triangle problems, vector geometry, physics, computer graphics, and signal analysis whenever an angle must be recovered from a cosine ratio or dot product.
Recover an angle from a known adjacent/hypotenuse ratio.
Find the angle between vectors using the dot product.
Convert cosine-based data into angular form.
Precision and rounding
The calculator rounds both radians and degrees to six decimal places. That is precise enough for most classroom, engineering, and graphing tasks while staying easy to read.
Outputs are rounded, not truncated.
Special values may display exact-looking decimals.
Small differences can occur if you round intermediate values by hand.
Common mistakes to avoid
Students often confuse inverse cosine with reciprocal cosine or forget the restricted principal range.
cos⁻¹(x) means inverse cosine, not 1/cos(x).
Do not enter values outside [-1, 1].
Remember that arccos returns the principal angle in the standard range.
FAQs
What is the domain of inverse cosine?
For real-number outputs, the domain is all x values from -1 to 1 inclusive.
What is the range of inverse cosine?
The principal range is 0 to π radians, or 0° to 180°.
Why is arccos(0) equal to 90°?
Because the cosine of 90° is 0, and 90° lies in the principal inverse cosine range.
Is cos⁻¹(x) the same as 1/cos(x)?
No. cos⁻¹(x) means inverse cosine, while 1/cos(x) is the secant of the original angle.
Why can’t I enter 2?
Because no real angle has cosine equal to 2. Real cosine values stay between -1 and 1.
Does the calculator show radians and degrees?
Yes. It reports the principal inverse cosine value in both units.
Can the result ever be negative?
Not for the principal inverse cosine range used here. Outputs stay between 0 and π radians.
What is arccos(-1)?
It is π radians, which is 180 degrees.
Where is inverse cosine used in practice?
It is used in trigonometry, vector angle calculations, engineering, physics, and graphics whenever an angle must be recovered from a cosine value.