Unit Circle Lab
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Inverse Cosine Calculator

This inverse cosine calculator finds arccos(x), the angle between 0° and 180° whose cosine is x, in degrees and radians. It returns exact values for special inputs, checks the domain, and draws the angle on the unit circle.

try

accepts 0.25 · −1/2 · √3/2 · sqrt(2)/2 · −1

xy120°

arccos(−1/2)

120° 2π/3 rad Principal value, in [0°, 180°]
Degrees
120°
Radians
2π/3
Principal range
[0°, 180°] · [0, π]
Solutions in [0°, 360°)
120° and 240°
General solution
θ = ±120° + 360°·k

Steps

rounded to 3–4 decimals
  1. Check the domain: arccos(x) needs −1 ≤ x ≤ 1, and x = −1/2 is inside it.
  2. Look for the angle θ in the principal range [0°, 180°], which is [0, π] in radians, with cos θ = −1/2.
  3. That is a special angle: cos 120° = −1/2, and 120° is in the range. So arccos(−1/2) = 120° = 2π/3.
  4. Cosine repeats, so arccos gives only one answer. All solutions of cos θ = −1/2: θ = ±120° + 360°·k.

How it works

Domain, range, then the angle.

  1. 01 · check

    Check the domain

    Cosine is an x-coordinate on the unit circle, so arccos only accepts values from −1 to 1.

  2. 02 · find

    Find the angle in range

    arccos returns the angle between 0° and 180°, on the top half of the circle, whose cosine is x.

  3. 03 · extend

    Get the other solutions

    The bottom half has a mirror angle, −θ. The panel lists both and the general solution.

§1What arccos means

Cosine turns an angle into a number: cos 60° = 1/2. Inverse cosine turns the number back into an angle: arccos(1/2) = 60°, which is π/3 in radians.

On the unit circle, cos θ is the x-coordinate of the point. arccos(x) asks “at what angle is the point x units to the right (or left) of the origin?” The dashed vertical line in the diagram marks that x-coordinate, and the cyan segment is the cosine.

§2Domain and range

The domain is [−1, 1]: the circle never goes farther than 1 unit left or right of the origin, so there is no angle whose cosine is 2 or −1.5.

The range is [0°, 180°], or [0, π]. Cosine gives the same value for θ and −θ, so the inverse has to pick one. It picks the top half of the circle, where every x-coordinate from 1 to −1 appears exactly once. Unlike arcsin, arccos never returns a negative angle.

§3Exact values of arccos

Inverse cosine of the special values
xarccos x (deg)arccos x (rad)
10°0
√3/230°π/6
√2/245°π/4
1/260°π/3
090°π/2
−1/2120°2π/3
−√2/2135°3π/4
−√3/2150°5π/6
−1180°π

Negative inputs give angles in Quadrant II, between 90° and 180°. In general arccos(−x) = 180° − arccos(x): arccos(−1/2) = 180° − 60° = 120°.

§4All solutions of cos θ = x

Each vertical line crosses the circle twice: at θ₀ = arccos(x) on the top half and at −θ₀ on the bottom half. With full turns added:

θ = ±θ₀ + 360°·k

cos θ = −√2/2. θ₀ = 135° → θ = 135° + 360°·k or −135° + 360°·k

In [0°, 360°) those are 135° and 225°. For x = 1 or x = −1 the two points meet, and there is only one solution per turn (0° or 180°).

§5arcsin x + arccos x = 90°

For any x between −1 and 1, arcsin x + arccos x = 90° (π/2). For example, arcsin(1/2) = 30° and arccos(1/2) = 60°, which add to 90°. This comes from cofunctions: the sine of an angle is the cosine of its complement. You can check it with the inverse sine calculator.

Inverse cosine is also the step that finishes the law of cosines: once you know cos C from three sides, C = arccos of that value. Because the range is 0° to 180°, it handles obtuse angles correctly, which arcsin cannot.

§6Common mistakes

  • Expecting arccos(cos θ) to give back θ for every θ. arccos(cos 300°) = arccos(1/2) = 60°, because 300° is outside the range.
  • Giving a negative principal value. arccos is never negative; −60° is a solution of cos θ = 1/2 but not the arccos.
  • Confusing cos⁻¹ x with 1/cos x, which is secant.
  • Using radian mode when the answer should be in degrees (arccos(0) is 90 in degrees but about 1.5708 in radians).

§7FAQ

What is arccos(0)?

90°, or π/2: the top of the circle has x-coordinate 0.

Why is arccos(2) undefined?

No point on a circle of radius 1 has an x-coordinate of 2, so no angle has cosine 2.

Is arccos the same as sec?

No. Secant is 1/cos x. Inverse cosine undoes cosine and returns an angle.

Can I enter decimals?

Yes. Decimals like 0.25 work and give a decimal angle. Exact forms like −√3/2 give exact answers such as 150° and 5π/6.