Question

The following two sets of parametric functions both represent the same ellipse, Explain the difference between the graphs.x = 3 cost and y = 8 sin tx= 3 cos 4t and y = 8 sin 4t

Answer 1

The graphs of both equations will have different slopes and periods

Step-by-step explanation:

Given:

`\begin{gathered} x=3\cos t\text{ and }y=8\sin t \\ x=3\cos4t\text{ and }y=8\sin4t \end{gathered}`

Recall that the equation of an eclipse is generally given as;

`(x^2)/(a^2)+(y^2)/(b^2)=1`

We can go ahead and express both equations in the above as seen below;

`\begin{gathered} x^2=3^2\cos^2t\text{ and }y^2=8^2\sin^2t \\ (x^2)/(3^2)+(y^2)/(8^2)=\cos^2t+\sin^2t \\ (x^2)/(3^2)+(y^2)/(8^2)=1 \\ OR \\ x^2=3^2\cos^24t\text{ and }y^2=8^2\sin^24t \\ (x^(2))/(3^(2))+(y^(2))/(8^(2))=\cos^24t+\sin^24t \\ (x^2)/(3^2)+(y^2)/(8^2)=1 \end{gathered}`

So we can see that both equations represent the same eclipse.

Recall the below sine function;

`y=a\sin(bx-c)+d`

where;

a = amplitude

2pi/b = period

If we compare the given equations, we can see that x = 3 cos t and y = 8 sin t, have 3 and 8 as amplitudes respectively and pi as period.

While x= 3 cos 4t and y = 8 sin 4t have 3 and 8 as amplitudes respectively and pi/4 as period.

If we express the equations as linear functions we'll have;

For x = 3 cos t and y = 8 sin t;

`\begin{gathered} (y)/(x)=(8\sin t)/(3\cos t) \\ y=(8)/(3)x\tan t \end{gathered}`
`\begin{gathered} (y)/(x)=(8\sin t)/(3\cos t) \\ y=(8)/(3)\tan t*x \end{gathered}`

For x = 3 cos 4t and y = 8 sin 4t

`\begin{gathered} (y)/(x)=(8\sin4t)/(3\cos4t) \\ y=(8)/(3)\tan4t*x \end{gathered}`

We can see that both equations have different slopes