In order to solve problems which require a sinusoidal model, it is necessary to

A typical problem requiring a sinusoidal model in which period,

frequency and

amplitude are important is a relationship between time and some other data. In many situations, this involves motion which repeats or oscillates. We are given some information about

data values that repeat over a certain interval or period of time, or we are given information about the position of an object that varies sinusoidally.

Suppose a particle moves along the x-axis. Its position (x-coordinate) at any time

**t** seconds where

**t** is greater than or equal to zero is given by

. (a) What is the position of the particle at time t = 2.3 seconds? (b) What are the amplitude, period and

frequency of this motion? (c) What is the smallest value of x that the particle reaches during its motion?

(a) To find the position of the particle at t = 2.3 we

evaluate

Note that this tell us the x-coordinate of the particle at t = 2.3.

.

.

From this we see that

Since

we can determine that the period is

T = 2 seconds.

(c) The maximum distance this particle moves can be seen easily if we note that at time

**t** = 0 the particle is at the coordinate x = 0. We call this the stable (equilibrium) position of the particle since it moves to the left and right of this position, which acts as a center of the motion. Since the

amplitude of the motion is 2, the particle moves from the origin at most a distance of 2. This means the smallest value of x that the particle reaches is x = -2. The particle moves back and forth between the x-coordinates -2 and +2 in a period of 2 seconds. A

graph of the position of this particle is shown below over a 10 second time interval.

Remember that the calculator uses X instead of t. So the

expression Y1=2sin(

pX) really represents Y1=2sin(

pt) or the values for our

function s(t). That is, the values of Y1 are the x-coordinates of the particle's position as it moves along the x-axis. Notice that at time t = 2.3 the particle's approximate position, or x-coordinate, is 1.618.