What’s the difference between an inverse and a reciprocal?

A ‘reciprocal’ means to divide 1 by the original function, for example, ##1/sin(x)##, but an ‘inverse’ of a trig function is an operation which determines the angle that led to a particular ratio.
In everyday life inverse and reciprocal are sometimes used interchangeably. In all fields of math, however, each means something completely different from the other.
A reciprocal refers to ‘1 divided by a number’, or ##1/sin(x)## for example. Since the functions ##(sintheta, costheta, tantheta)## each refer to a ratio of two sides, the reciprocal simply ‘flips’ the ratio over, and we give it a new name. Sine becomes Cosecant, Cosine becomes Secant, and Tangent becomes Cotangent.
But an inverse in trig refers to the determination of an angle which gave us a particular trig ratio. In a right triangle, suppose the side opposite the angle is ##3##” long and the hypotenuse is ##6##” long. We would find the sine value by dividing the opposite over hypotenuse, so ##sin(x) = 3/6 -> sin(x)=0.5##. We could then use the ‘inverse’ sine of that ratio to tell us what angle led to that ratio.
##sin^-1(0.5) = 30^@##
That means that any right triangle where the opposite side is half as long as the hypotenuse must have a ##30^@##angle.