It is just a name invented by mathematicians.
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No. The inverse of the secant is called the arc-secant. The relation between the secant and the cosecant is similar to the relation between the sine and the cosine - they are somehow related, but they are not inverse functions. The secant is the reciprocal of the cosine (sec x = 1 / cos x). The cosecant is the reciprocal of the sine (cos x = 1 / sin x).
One is the inverse of the other, just like the arc-sine is the inverse of the sine, or division is the inverse of multiplication.
An arc whose measure is less than 180 degrees is called a Minor Arc.
Semi-circle.
A line between 2 vertices in Graph theory is called an edge or an arc, although arc is usually used to denote a directed edge.