• use basic graphing skills for sine and cosine
• know how to find amplitude, period, and frequency
• solve equations requiring the use of inverse sine and cosine: arccos(x) or cos^-1(x) and arcsin(x) or sin^-1(x)

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.

 

(b) The amplitude is given as 2, the leading coefficient in the original equation

 

.

 

The frequency can be found by rewriting the expression

 

.

 

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.