Question

Please help, I am asked to sketch the curve by eliminating the parameter and indicate the direction of increasing t for the following parametric equations. (click to see).

Answer 1

x=2cos(t)
y=5sin(t)
0 (greater than or equal to ) t (greater than or equal to) 2pi

How do I determine what values for t are appropriate in this case?

Answer 2

Where is the picture?

Answer 3

There is no picture, I have to sketch it.

Answer 4

oh ok lol.

Answer 5

You see, there is a \(\cos(t)\) in the first coordinate, there is a \(\sin(t)\) in the second coordinate. The idea of the exercise is to forget about the \(t\) for an instant and look at a relation between \(x\) and \(y\). Do you know the fundamental formula involving \(\cos\alpha\) and \(\sin\alpha\)?

Answer 6

I know of addition and double-angle formulas.

Answer 7

and half-angle formulas

Answer 8

No, you must write the formula \(\cos^2t + \sin^2t=1\) in terms of \(x\) and \(y\). After doing so you should recognize the equation of a known curve.

Answer 9

\(\cos(t)=\frac x2\) and \(\sin(t)=\frac x5\) right?
Write the formula and you will obtain the equation of an ellipse. It is easy to sketch (choose a few values for \(t\) to obtain some points you are sure of).

To know the direction, choose \(t=0\) and then \(t=\pi/2\) for example, and look the direction in which the point has "moved".

Answer 10

I mean, \(\sin(t)=\frac y5\)

Answer 11

$$
x=2cos(T) \implies cos(T)=\cfrac{{(x)}}{2}\\
y=5sin(T) \implies sin(T)=\cfrac{{(y)}}{5}\\
\text{now keeping in mind the pythagorean identity } cos(\theta)^2+sin(\theta)^2=1\\
\cfrac{(x)^2}{2^2}+\cfrac{(y)^2}{5^2}=1 \implies \cfrac{(x-0)^2}{2^2}+\cfrac{(y-0)^2}{5^2}=1
$$