Angular Distance is the true metric version of cosine distance

DEFINITION

Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors in three-dimensional space, or the central angle subtended by the radii through two points on a sphere.
— https://en.wikipedia.org/wiki/Angular_distance

Since it is the angle between two vectors it can be derived from the Cosine Similarity is the cosine of the angle between two vectors.

When the vectors may be positive or negative:

$$d_{ang}=\frac{\arccos (d_{cos})}{\pi }$$

if the vector elements are always positive:

$$d_{ang}=\frac{2\arccos (d_{cos})}{\pi }$$

See also

• Cosine Similarity is the cosine of the angle between two vectors - angular distance is the metric-compliant conversion of cosine distance (i.e. it does not violate the triangle inequality)
• SimHash - h(x) = sign(wᵀx) - the factor $\theta$ in SimHash’s collision probability is exactly the numerator of angular distance
• Clustering is the task of grouping a set of objects - clustering algorithms that require a true metric (i.e. k-center) need angular distance rather than cosine for correctness guarantees
• A Hash Function Must Be Deterministic, Uniform, and Fast — FNV-1a Is One Example - angular distance is what LSH schemes for cosine similarity are actually preserving, in fact SimHash maps nearby vectors in angular space to the same bucket