Inverse Cosine Calculator (Arccos)
The inverse cosine of 0.5 is 60 degrees, or about 1.0472 radians. This inverse cosine calculator, also called an arccos or cos inverse calculator, converts a cosine ratio between -1 and 1 into its principal angle. It reports degrees, radians, and radians divided by pi. In a right triangle, the cosine ratio is adjacent side divided by hypotenuse. An adjacent side of 5 next to a hypotenuse of 10 gives a ratio of 0.5, and arccos(0.5) gives an angle of exactly 60 degrees. Because a hypotenuse is always the longest side, the ratio adjacent over hypotenuse always falls between -1 and 1, which is why arccos rejects any input outside that range. Arccos returns one principal answer between 0 and 180 degrees. That range is different from arctan, which returns answers between -90 and 90 degrees. Cosine only needs the wider 0 to 180 range because it is not periodic in the same half-turn sense as tangent, and every value from -1 to 1 maps to exactly one angle across a full half circle.
Quick answer
Cosine turns an angle into a ratio. Inverse cosine goes the other way, from ratio back to angle.
Angle
60 degrees
Radians
1.047198
Radians / pi
0.333333
What this tells you
- •Cosine turns an angle into a ratio. Inverse cosine goes the other way, from ratio back to angle.
- •Arccos only accepts numbers from -1 to 1, since cosine itself never produces a ratio outside that range.
- •Arccos always answers with an angle between 0 and 180 degrees, unlike arctan's -90 to 90 range.
- •Arccos of 1 is 0 degrees, arccos of 0 is 90 degrees, and arccos of -1 is 180 degrees.
- •A positive ratio gives an angle under 90 degrees, and a negative ratio gives an angle over 90 degrees.
- •The displayed degree angle is rounded to 4 decimal places, while the secondary degree and radian values are rounded to 6 decimal places.
How to Use
- 1Enter a ratio between -1 and 1, like adjacent divided by hypotenuse from a right triangle.
- 2Read the angle in degrees as the main result.
- 3The radians value and its multiple of pi are shown below for math homework that wants exact form.
- 4For a triangle side problem, divide the known adjacent length by the hypotenuse length before entering it.
- 5Remember the input must stay between -1 and 1. Anything outside that range is not a valid cosine ratio.
- 6Check whether the problem calls for arccos, arcsin, or arctan based on which two sides you know.
- 7Use the radians-over-pi result as a decimal multiple of pi, not automatically as an exact fraction.
How It Works
Formula
angle = arccos(x), where cos(angle) = x, for -1 <= x <= 1Arccos answers which principal angle has cosine x. Since cos(60 degrees) = 0.5, arccos(0.5) = 60 degrees. The code first uses Math.acos(x), which returns radians in the interval from 0 to pi inclusive. It converts radians to degrees with radians x 180 / pi. For an adjacent side of 5 and a hypotenuse of 10, x = 5 / 10 = 0.5. Math.acos(0.5) is exactly pi/3 radians, or about 1.047198 radians, and multiplying by 180 / pi gives exactly 60 degrees. The output rounds degrees and radians to 6 decimal places, radians divided by pi to 6 decimal places, and the formatted degree result to 4 decimal places. Any input below -1 or above 1 returns null because no real angle has a cosine outside that range.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Arccos of 1
A cosine ratio of 1 means the adjacent side equals the hypotenuse, which only happens at a 0 degree angle.
Arccos of 0
A zero cosine ratio means the adjacent side has shrunk to zero. Math.acos(0) returns pi/2 radians, which is 90 degrees.
Arccos of 0.5
An adjacent side of 5 over a hypotenuse of 10 gives a ratio of 0.5, and arccos(0.5) is exactly 60 degrees.
Arccos of a negative ratio
Negative ratios return angles greater than 90 degrees. Arccos(-0.5) is exactly 120 degrees.
Arccos of -1
A cosine ratio of -1 is the largest possible angle in the principal range, exactly 180 degrees or pi radians.
Common Arccos Values
Inputs with clean angle answers, in degrees and radians.
| x | arccos(x) degrees | arccos(x) radians |
|---|---|---|
| 1 | 0 | 0 |
| 0.866 | 30 | pi/6 |
| 0.5 | 60 | pi/3 |
| 0 | 90 | pi/2 |
| -0.5 | 120 | 2pi/3 |
| -0.866 | 150 | 5pi/6 |
| -1 | 180 | pi |
Why arccos uses a 0 to 180 degree range
Cosine measures the ratio between an adjacent side and a hypotenuse in a right triangle, and that ratio ranges from -1 to 1 across a half circle from 0 to 180 degrees. Arccos is defined to invert exactly that half circle, so every input from -1 to 1 maps to one and only one output between 0 and 180 degrees.
This is a key difference from arctan, which returns angles between -90 and 90 degrees, and from arcsin, which returns angles between -90 and 90 degrees as well. Arccos is the outlier of the three because cosine values repeat in a way that only a 0 to 180 window can invert without ambiguity. Picking the right inverse function for a triangle problem depends on which sides or ratios you already know.
Radians divided by pi can reveal familiar angles. For x = 0.5, the result is exactly one third because pi/3 divided by pi equals 1/3. Most inputs do not produce a simple exact fraction, so a displayed decimal such as 0.583333 should be treated as rounded unless it matches a known value.
Common mistakes
- Entering a value outside -1 to 1. Cosine never produces a ratio beyond that range, so arccos of 1.5 or -2 has no real answer and returns null.
- Reading the result in the wrong unit. 60 degrees and about 1.047 radians are the same angle, and calculators in the wrong mode cause most arccos errors.
- Assuming arccos shares arctan's -90 to 90 range. Arccos always returns an angle between 0 and 180 degrees instead.
- Confusing arccos with 1/cos. The -1 in cos-1 means inverse function, not reciprocal. The reciprocal of cosine is secant.
- Dividing the wrong two sides. Cosine is adjacent over hypotenuse, not adjacent over opposite or opposite over hypotenuse.
- Entering an angle when the tool expects a cosine ratio. To evaluate cosine from an angle, use a trigonometry calculator instead.
- Forgetting that a negative ratio is valid input. Arccos(-0.5) correctly returns 120 degrees, not an error.
- Assuming the radians-over-pi decimal is an exact fraction. It is rounded to 6 decimal places unless the value happens to map cleanly.
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