If the tan A = 0.9063, find m ∠A to the nearest degree.

You would use the arctan of 0.9063 (it is the inverse of the tangent, so it finds the angle's measurement):

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\mathtt{0.906\: \!3}}\right)} = {\mathtt{42.186\: \!016\: \!222\: \!452^{\circ}}}$$

So angle A is about 42 degrees.