Arccos Calculator
What the Arccos Calculator returns
Given the cosine of an angle, the Arccos Calculator recovers the angle itself. Robotics engineers extract joint rotations from dot products, physics students find the phase angle between two waves, and trigonometry learners finish law of cosines problems with it. Everything rests on the fact that arccos undoes the cosine function.
Inverting cosine with a worked number
Arccos(y) returns the unique angle between 0 and 180 degrees whose cosine equals y. The anchor case is arccos(0.5) = 60 degrees, because cos(60 degrees) = 0.5 in the 30-60-90 triangle. Similarly arccos(0.7071) gives 45 degrees, and arccos(0) = 90 degrees.
Data scientists borrow the same field for similarity work. If two product rating vectors produce a cosine similarity of 0.8, entering 0.8 into the Cosine field returns the angle between them: arccos(0.8) = 36.87 degrees, or 0.6435 radians in the other unit. Identical vectors share a similarity of 1, and the calculator correctly answers 0 degrees, while unrelated vectors near 0 steer close to 90 degrees. For contrast, a weaker match of 0.6 opens the angle to 53.13 degrees, showing how quickly distance grows as agreement fades. Converting a similarity score into degrees this way turns an abstract statistic into a geometric picture a whole team can discuss.
Input limits, range, and units
Place your value in the Cosine field and press Calculate; only decimals from -1 to 1 are accepted. Choose degrees or radians for the output. Answers always land between 0 and 180 degrees (0 to pi radians), so arccos(0.5) returns 60 degrees while arccos(-0.5) returns 120 degrees.
Frequently Asked Questions
What is arccos of 0.5?
Arccos(0.5) equals 60 degrees, or pi/3 radians, because cos(60 degrees) = 0.5 exactly. The number comes straight from the 30-60-90 triangle, whose adjacent leg is half the hypotenuse.
Which inputs does arccos accept?
Only values between -1 and 1 inclusive, because cosine never leaves that interval. Thus arccos(0.5) = 60 degrees is fine, while arccos(2) has no mathematical answer.
Why does arccos span 0 to 180?
Cosine falls steadily from 1 at 0 degrees to -1 at 180 degrees without repeating, so every input between -1 and 1 matches exactly one angle in that interval.
What is arccos of negative 0.5?
It equals 120 degrees, since cos(120 degrees) = -0.5. Negative inputs always produce obtuse angles between 90 and 180 degrees, and 120 degrees fits that principal range.
How does arccos help in the law of cosines?
After computing cos(C) = (a^2 + b^2 – c^2)/(2ab), apply arccos to obtain angle C. For a = 5, b = 6, c = 7, cos(C) = 0.2, so C measures about 78.46 degrees.