Find the smallest interval for parametric equations

Question

$\mathcal{G}$ is the graph of parametric equations $\begin{align*} x = \cos(4t), y = \sin(6t). \end{align*}$.

Find the length of the smallest interval $I$ such that the graph of the parametric equations for all $t\in I$ produces the entire graph $\mathcal{G}$.

I am not very good with parametric equations so I am not sure where to begin. Any hints?

abel · Accepted Answer · 2015-02-04T01:58:15.690

1

if you look at the parametric graph on the intervals $0 \le t \le \pi/4$ it does an arc staring at $(1,0)$ to $(-1,-1)$ on the next $\pi/4 \le t \le \pi/2$ it retraces that arc back to $(1,0)$ for $\pi/2 \le t \le 3\pi/4$ it does an arc that is the reflection of the old arc traversed for $0 \le t \le \pi/4 $ on the $x$-axis. and finally for $3\pi/4 \le t \le \pi$ it is back to $(1,0)$ so the period is $\pi.$

edited Feb 04 '15 at 01:58

answered Feb 04 '15 at 01:23

abel

29,170

What reflection are you referring to by "reflection of the old"? – Mathy Person Feb 04 '15 at 01:32
If the period is $\pi$, then the length of the smallest interval is...$\pi$, too? – Mathy Person Feb 04 '15 at 01:57
yes @MathyPerson, the smallest interval is $\pi$ too. – abel Feb 04 '15 at 01:59
hmm.. I tried it and $\pi$ was incorrect. – Mathy Person Feb 04 '15 at 02:03
@MathyPerson, the graph is produced twice in that interval $0 \le t \le \pi.$ you can produce the graph in the shorter time $\pi/4 \le t \le 3\pi/4.$ the shortest interval has length $\pi/2.$ – abel Feb 04 '15 at 02:06

score 1 · Answer 2 · answered Feb 04 '15 at 02:07

1

According to M.G., you may want to have the x and y to repeat themselves, but they may have passed different number of periods, that is to say: $$ \begin{cases}4t=n_1T\\6t=n_2T\end{cases} $$ therefore： $$ \frac{n_1}{n_2}=\frac{2}{3} $$ $n_1=2$ is the minimum number for the equation, and $t=(1/4)(2)(2\pi)=\pi$ is the solution.

answered Feb 04 '15 at 02:07

PdotWang

749

Best answer, since it is general! Is it correct to say that when looking for a period of parametric equations, we can find a period of each equation and then find the least common multiple - that is the period of the parametric equation? – Martinsos Apr 13 '17 at 12:49

score 0 · Answer 3 · answered Feb 04 '15 at 01:13

0

Hint:

Remember that $\cos$ and $\sin$ are periodic functions with period $T=2\pi$.

answered Feb 04 '15 at 01:13

Ángel Mario Gallegos

19,882

Is it $[\frac{\pi}{3}, \frac{\pi}{2}]$? – Mathy Person Feb 04 '15 at 01:18

Find the smallest interval for parametric equations

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