How this calculator works
Cosine decreases from one to minus one as the angle runs from 0° to 180°. That interval defines the principal arccosine branch, so all returned principal angles are non-negative. A negative ratio therefore gives an obtuse angle. Reflection across the horizontal axis gives the other full-turn cosine solution. At ratios 1 and -1 the two reflected solutions coincide.
Formula
y=arccos x means cos y=x with 0≤y≤π. cos θ=x has θ=y+2kπ or θ=-y+2kπ.
Worked example
arccos(0.5)=60°=π/3, with 300° as the other first-turn cosine solution. arccos(-0.5)=120°, and arccos(-1)=180°.
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Frequently asked questions
Can arccos return a negative angle?
The principal arccosine branch used here is 0° through 180°, so its result is never negative.
Why does a negative cosine give an obtuse angle?
On the upper half of the unit circle, negative horizontal coordinates occur beyond 90°.
How is this used with three triangle sides?
The law of cosines forms a ratio from the squared sides, then arccos recovers the opposite angle. The side lengths must first form a valid triangle.