In other areas of mathematics, an exponent of $\displaystyle-{1}\usepackage{cmbright}\Huge\$ does indicate a reciprocal. However, when it is used with a trigonometric function, it indicates the inverse of the function. The inverse switches the input and output of the original function. It is correct that this has something to do with the sine of $\displaystyle{{30}}^{\circ}\usepackage{cmbright}\Huge\$ , but no reciprocals are involved.