>> The plot below is a Lissajous figure created with the parametric equations x
of t equals a cosine bt, and y of t equals c, sin dt.
We want to find possible positive values for a, b, c, and d. To begin we should be familiar
with our basic cosine function graphed, here in red,
and the basic sine function graphed here in black.
Notice how looking at the parametric equations the x values are controlled
by the cosine function and the y values are controlled by the sine function.
Both cosine and sine, have a maximum function value of 1,
and the minimum function value of negative 1.
So, by analyzing the x coordinates of our figure, we should be able to determine a,
where the absolute value of a would be the amplitude of our cosine function
and by analyzing the y coordinates of our figure, we should be able to find c,
where the absolute value of c would be the amplitude of our sine function.
We'll begin by determining a, by analyzing the x values of our figure, more specifically,
the maximum x value and the minimum x value.
Looking at our figure, we project this figure
on to the x axis notice how the maximum x value is positive 6,
and the minimum x value is negative 6.
And, therefore, since you want to use a positive value for a, a would have to be 6,
meaning the amplitude of our cosine function would be 6.
So, to begin, we know that x of t, must equal 6, times cosine of bt, and now,
we'll analyze the y coordinates of our figure to determine c where the absolute value
of c would be the amplitude of the sine function.
So, if we project our figure on to the y axis,
notice how the maximum function value is positive 5,
and the minimum function value would be negative 5.
And once again, because you want to use a positive value for c, c would be positive 5.
So, now we know that y of g, equals 5, times sine, dt.
Remember the coefficient of t would affect the period of the cosine and sine functions.
Where if we have y equals sine b theta or y equals cosine b theta, the period would be equal
to 2 pi divided by b, when the frequency would be the reciprocal of the period.
Now, b and d are not unique even though their ratio will be.
To simplify things, let's assume we want the entire figure to be traced from t equals zero
to t equals 2 pi radius [writing].
This doesn't have to be true but it will simplify things slightly.
So, what we'll do is trace out this figure and determine how many cycles of the cosine function
and how many cycles of the sine function it will take which will help us determine b
and d. Before we do this though, let's find the point in our figure when t equals zero.
Notice when t equals zero, x of zero would be six times cosine zero which would be 6,
and y of zero would be 5, times sine zero which would be zero.
So, when t equals zero, we'd be at this point on the figure and the orientation,
as t increases would be in this direction here.
And that will determine how many cycles
of our cosine function it will take to trace out our figure.
Before we do this though, we take a look at the basic cosine function.
Notice how it starts at a maximum and then when it returns to that maximum,
we have one complete cycle of our cosine function.
So, here we're starting at positive 6, every time we return to positive of six,
it represents one cycle of our cosine function.
Let's go ahead and begin tracing.
[ Drawing ]
Notice here were turning back to positive 6, this represents one complete cycle
of our cosine function and we keep going [drawing], we're back to positive 6 again,
that's two complete cycles [drawing], and we're back to positive 6 a third time, and therefore,
it takes three complete cycles of our cosine function to trace our figure.
And since you want to trace the figure from t equals zero to t equals 2 pi radians that means
that the period of our cosine function would have to be 2 pi divided by 3, and therefore,
b, the coefficient of t must be 3.
We can also say the frequency would be three cycles per 2 pi radians.
So again, b is equal to 3.
So, now we know that x of t, equals 6 times cosine of 3t.
And now we'll trace the entire figure again and determine how many cycles
of the sine function it would take to trace our figure.
But, before we do this, let's look at the basic sine function.
Notice that the sine function starts at zero, and then it has to return to zero, not once,
but twice for one complete cycle.
So, the y value would have to return to zero twice
to represent one complete cycle of the sine function.
So, let's go ahead and start tracing.
They y value starts at zero, here it returns to zero once, but it has to return
to zero twice for one complete cycle.
So, we keep tracing [writing], we're back to zero a second time here.
So, that represents one complete cycle of our sine function.
Keep tracing [writing], at zero once, and at zero twice.
So, it takes two complete cycles of our sine function to trace this figure, and therefore,
if we want to use t equals zero to t equals two pi radians to trace our figure,
the period would have to be 2 pi divided by 2, and therefore, b, or in this case d,
the coefficient of t, would have to be 2.
And therefore, y of t [writing] equals 5 sine two t. Now as I mentioned earlier,
b and d are not unique but their ratio is.
As long as the cosine function completes three cycles,
every time our sine function completes two, we would trace out our figure but,
the interval for t, would change.
But, I think using these t values does help simplify the process slightly.
I hope this was helpful. ------------------------------246f76f3b2e6--