Cos Inverse Calculator (Arccos)
Enter a cosine value between -1 and 1 to find the principal angle whose cosine equals that value. This inverse cosine calculator returns the result in degrees, radians, and pi-radians simultaneously. You also get a show-your-work steps panel, a labeled unit-circle curve visual, and the exact symbolic form whenever it exists.
What is the inverse cosine (arccos)?
The inverse cosine function, written arccos(x), cos⁻¹(x), or acos(x), answers the question: what angle has a cosine equal to x? Because cosine is periodic, there would be infinitely many answers, so arccos is restricted to the principal range: it always returns an angle in [0, 180 deg] (or equivalently [0, pi] radians). This unique-answer range is called the principal value. For example, both 60 deg and 300 deg have a cosine of 0.5, but arccos(0.5) returns only 60 deg. The input x must lie in [-1, 1] because the cosine of any real angle never leaves that interval. Inputs outside this domain have no real inverse and the calculator returns no result.
How to use this calculator
Type any cosine value between -1 and 1 in the input field. The calculator instantly returns the angle in three equivalent forms: degrees (the most human-readable form, 0 to 180), radians (0 to pi), and pi-radians (0 to 1, expressed as a multiple of pi). For unit-circle values like 0, 0.5, -0.5, sqrt(2)/2, sqrt(3)/2, and 1 and -1, the exact symbolic radian form is shown as well. The steps panel beneath the result traces the computation, and the curve chart shows where your point sits on the full arccos graph. Use the primary output unit selector to choose whether degrees or radians appear at the top of the result card.
The arccos formula and domain
Formally, y = arccos(x) means x = cos(y) with y in [0, pi]. The domain is -1 to 1 and the range is 0 to pi radians (0 deg to 180 deg). The function is strictly decreasing: arccos(-1) = 180 deg, arccos(0) = 90 deg, arccos(1) = 0 deg. To convert between the two common output units, use: degrees = radians * (180 / pi) and radians = degrees * (pi / 180). The complementary relationship between inverse sine and inverse cosine is: arcsin(x) + arccos(x) = pi/2 for all x in [-1, 1]. This identity means that if you know arccos(x), you get arcsin(x) for free: just subtract from 90 deg.
Applications of inverse cosine
Arccos appears across many practical problems. In geometry, it gives the angle between two vectors from the dot-product formula: angle = arccos(A . B / (|A| |B|)). In navigation and surveying it converts horizontal and vertical distances into bearing angles. In physics and engineering it appears in the law of cosines for finding angles in any triangle: angle C = arccos((a^2 + b^2 - c^2) / (2ab)). In computer graphics it recovers rotation angles from transformation matrices. In signal processing and oscillations, phase differences between sinusoidal waveforms are calculated with arccos. Because the function output is always between 0 and 180 degrees, results outside that range indicate the need for the full four-quadrant version (atan2-style handling) or a different inverse trig function.
Common arccos values (unit circle)
| x = cos(angle) | Degrees | Radians | Exact form |
|---|---|---|---|
| -1 | 180 | 3.14159 | pi |
| -sqrt(2)/2 | 135 | 2.35619 | 3*pi/4 |
| -1/2 | 120 | 2.09440 | 2*pi/3 |
| 0 | 90 | 1.57080 | pi/2 |
| 1/2 | 60 | 1.04720 | pi/3 |
| sqrt(2)/2 | 45 | 0.78540 | pi/4 |
| sqrt(3)/2 | 30 | 0.52360 | pi/6 |
| 1 | 0 | 0 | 0 |
Exact values for angles that appear in standard trigonometry and geometry.
Frequently asked questions
What is the difference between arccos and cos⁻¹?
They are the same function. arccos(x), cos⁻¹(x), and acos(x) are all notations for the inverse cosine. The superscript -1 in cos⁻¹ means "inverse function", not "one divided by cosine" (which would be the secant). All three notations appear in textbooks and calculators, so recognising them is important.
Why must x be between -1 and 1?
The cosine of any real angle is always between -1 and 1 (inclusive). Consequently the inverse cosine has no real output for values outside that interval. For example, arccos(2) does not exist in the real numbers, though a complex-number result can be computed. This calculator works only with real inputs and returns no result for |x| > 1.
Why does arccos always return an angle between 0 and 180 degrees?
The cosine function is not one-to-one over its full domain, so to define a proper inverse, mathematicians restrict the output to a single "principal" interval. The standard choice is [0, 180 deg] (or [0, pi] radians), because every value of x in [-1, 1] corresponds to exactly one angle in that range. This is analogous to how arcsin is restricted to [-90 deg, 90 deg].
How do I convert arccos from radians to degrees?
Multiply the radian result by 180/pi. For example, arccos(0.5) = pi/3 radians. Multiplying by 180/pi gives 60 degrees. The calculator shows both automatically, so you never need to convert manually.
What is the relationship between arccos and arcsin?
For any x in [-1, 1], arcsin(x) + arccos(x) = 90 degrees (pi/2 radians). This complementary identity follows from the geometry of a right triangle: if arccos(x) gives one acute angle, arcsin(x) gives the other (since the two acute angles in a right triangle always sum to 90 degrees). The calculator shows this as the "complementary angle" output.
How do I find an angle in a triangle using the inverse cosine?
Use the law of cosines: for a triangle with sides a, b, c, the angle C opposite side c is C = arccos((a^2 + b^2 - c^2) / (2ab)). Compute the fraction inside the brackets first, then enter it into this calculator to get angle C in degrees or radians.