One skill that engineers and engineering students must develop is the ability to translate one mathematical model into another quickly and with fluency.

For example, the following equations describe the motion of a vibrating spring:

.

In most engineering applications, this formula is translated into a simpler form, using basic trigonometric identities. The new formula is

in which the amplitude of the motion is the constant A and the "angle" acts as a shift in time. Use some basic trigonometric identities to show that the two formulas are connected by the relationships

.