Hello Craig,

For the trigonometric functions sin and cos the constants in the equation determine the magnitude, frequency and phase. For the equation y = 3 * sin( pi*x + pi/2) the amplitude is 3, the frequency is pi and the phase shift is 1/2.

A generalized equation would be:

(Eq 1) y = Amp * sin( Freq*(x - Phase)) (If you're wondering about the minus sign see my note below)

Another form of the equation is (as in the case of your problem):

(Eq 2) y = Amp * sin( Freq*x + ZZZ)

In this case you must calculate the phase shift by dividing ZZZ by Freq, i.e. ZZZ/Freq = (π/2)/π = 1/2 (for my example above).

You can get a better grasp of this by using one of the web sites that shows the graph of a function you enter, Fooplot for instance.

You can enter several functions and see the differences in the graphs.

Try entering the following and note how the plot changes:

2*sin(pi/3*x+pi)

2.5*sin(pi/3*x+pi)

2*sin(pi/2*x+pi)

2*sin(pi/3*x+pi/2)

You can think of period as simply the inverse of frequency. If the frequency is 3 the period is 1/3. If the frequency is 3/2 the period is 2/3... BUT this is true only if the input to the sine function is in REVOLUTIONS. Since the input to the sine function is usually in RADIANS the result must be multiplied by 2π (because one full revolution is 2π radians). So the answers for the above are more correctly 2π*1/3 = 2/3π and 2π*2/3 = 4π/3.

The use of the word "period" in math and science is slightly different than other fields of study (history, for instance). This isn't a problem with the type of problem we're discussing, but it can be misunderstood in word problems.

When a math problem asks about "period" it means the interval of time between events. Don't confuse this with the interval of time during which measurements are taken!

As an example, to find the rate your heart beats you might count the number of beats for a "period" of one minute. This could be referred to as "the period of measurement", but "one minute" is NOT the correct answer for the problem! For the purposes of math and science, the word "period" means "the time interval between events." To find the correct answer I suggest you find the frequency first, and then invert the frequency to find the period.

If you counted 80 heart beats in one minute the frequency of heart beats is (obviously) 80 beats per minute. The period is then one eightieth (1/80) minutes per heart beat. If we use seconds instead of minutes for the unit of time we would get a different (but still correct) answer: 80 beats per 60 seconds is 80/60 = 8/6 = 4/3 = 1 1/3 beats per second. This is the frequency so the period is then the inverse: 60 seconds per 80 beats = 60/80 = 6/8 = 3/4 seconds per beat.

Note -- the minus sign in Eq 1 indicates the DIRECTION of the phase shift, left or right. If you haven't discussed this in class you probably don't need to worry about it.

Byron S.

10/12/14