x=2sin(3t) and y=3sin(1.5t), t ≥ 0 is a parametric representation of a closed curve. There is therefore, a value p for which if y(t)=f(x(t)) then y(t+p)=y(t) for all t ≥ 0. p is called the period of the function. Determine the period of this parametric curve. Please help (:

Question

anonymous · Answer

Recall the double angle formula for sine:
$$\sin2x=2\sin x\cos x$$
You can use this to eliminate $\sin1.5t$, leaving you with a factor of $\cos1.5t$.

anonymous · Answer

Okay. Do I do this to both x and y or would the equations now be $$x=2\sin 3t $$ and$$y=3\cos 1.5t$$. If so where do I go from here

anonymous · Answer

Just for $x$:
$$x=2\sin3t=2\bigg(2\sin\frac{3}{2}t\cos\frac{3}{2}t\bigg)=4\sin1.5t\cos1.5t$$
You may also want to recall the Pythagorean identity to get rid of the cosine:
$$x=4\sin1.5t\cos1.5t=4\sin1.5t\sqrt{1-\sin^21.5t}$$
Then, since $y=3\sin1.5t$, you can write
$$x=4\left(\frac{3\sin1.5t}{3}\right)\sqrt{1-\frac{9\sin^21.5t}{9}}\
x=\frac{4}{3}\left(3\sin1.5t\right)\sqrt{1-\frac{(3\sin1.5t)^2}{9}}\
x=\frac{4}{3}y\sqrt{1-\frac{y^2}{9}}$$
Note that using the positive root only gives you one half of the given parametric curve; including the negative root should give the other half.

I'm not sure about the period part of the question...

anonymous · Answer

Okay. Thank you. It helps a lot being able to see the equations written this way. Maybe I can pull the period from this

anonymous · Answer

A Mathematica solution and associated plots are attached.

anonymous · Answer

Thank you so much robtobey!! I appreciate it so much. This was so helpful!