The following two sets of parametric functions both represent the same ellipse. Explain the difference between the graphs.
x=3 cos t and y=8 sin t
x= 3 cos 4t and y=8 sin 4t
they have the same semi-axes; one just traces the curve faster than the other.
Well, it seems like someone got a bit trigger happy with the "cos" and "sin" functions here! Let me explain the difference between these two sets of parametric functions.
In the first set, x = 3 cos t and y = 8 sin t, both the x and y coordinates are determined by the same angle t. This means that as t varies, the coordinates (x, y) will trace out points on the ellipse. The parameters a = 3 and b = 8 determine the size and shape of the ellipse, but the values of t only affect the overall position of the ellipse in the coordinate plane.
Now, in the second set, x = 3 cos 4t and y = 8 sin 4t, we have 4t instead of just t as the angle for both the x and y coordinates. This means that the angle is changing four times as fast as in the first set. As a result, the points on the ellipse will be more tightly packed together along the circumference.
In simpler terms, the second set of functions produces a "squished" version of the ellipse. It's like taking the original ellipse and squeezing it inwards, making it more elongated.
So, to summarize, the difference between the two graphs lies in how quickly the angle is changing, resulting in a different spacing and shape of the points on the ellipse.
The difference between the graphs lies in the parameter used in the cosine and sine functions.
In the first set of parametric functions, x = 3 cos(t) and y = 8 sin(t), the angle "t" determines the position of the points on the ellipse. As "t" varies from 0 to 2π (or 0 to 360 degrees), the points trace out the entire ellipse once. The ellipse is stretched horizontally by a factor of 3 (the coefficient of cos(t)) and stretched vertically by a factor of 8 (the coefficient of sin(t)).
In the second set of parametric functions, x = 3 cos(4t) and y = 8 sin(4t), the angle "t" is multiplied by 4 in the cosine and sine functions. This means that the points on the ellipse complete four revolutions around the ellipse as "t" varies from 0 to 2π. The ellipse is compressed horizontally since the period of the cosine function (4t) is shorter, and it is compressed vertically since the period of the sine function (4t) is also shorter.
To summarize, the first set of parametric equations creates an ellipse that completes one revolution, while the second set of parametric equations creates an ellipse that completes four revolutions. The second set is compressed in both the horizontal and vertical directions compared to the first set.
To understand the difference between the graphs of these two sets of parametric functions, let's examine the components of each equation separately.
The first set of parametric functions is:
x = 3cos(t)
y = 8sin(t)
The second set of parametric functions is:
x = 3cos(4t)
y = 8sin(4t)
The key distinction lies in the argument inside the cosine and sine functions. In the first set, the argument is 't', whereas in the second set, the argument is '4t'.
The role of the argument is to determine the rate at which the parameter 't' progresses, consequently affecting the shape and orientation of the graph.
For the first set, with x = 3cos(t) and y = 8sin(t), as 't' ranges from 0 to 2π (a full revolution), x and y will cover one complete ellipse. The cosine and sine functions generate the points on the unit circle, and multiplying by 3 and 8 scales the ellipse along the x and y axes, respectively.
In the second set, with x = 3cos(4t) and y = 8sin(4t), the argument of '4t' causes the parameter to increase at a quadruple rate compared to the first set. As a result, x and y will go through four full revolutions as 't' moves from 0 to 2π. This makes the second graph appear more compact than the first one.
To summarize, the main difference between the graphs is the rate at which they complete revolutions due to the argument inside the cosine and sine functions. Changing the argument value affects the frequency of repetition and, in turn, alters the shape and compactness of the resulting ellipse.