In my math book there are these parametric equations of the orbit/trajectory of a point $P$: $$\begin{cases} x(t)=\sin t\\ y(t)=\sin 4t\\ \end{cases} $$ along with a graph/picture of it on the $y$ and $x$ axis. Could anyone explain why it follows the path it does?
I unfortunately cant plot it on desmos or show a picture, so if anyone could do this, that would be great.
I thought about it, and since the x values are given by $sin(t)$, does this mean that the trajectory of $P$ first goes to the right, but after $sin(\pi)$ the $x$ values suddenly get negative, and the trajectory switches to the left of the $y$ axis?
So my question is in which direction does the orbit exactly go?
Thanks in advance
The figure you're looking at is determined by looking at an $x$-$y$ plane and by drawing out a path $(x(t),y(t))$ that depends on a parameter $t\in \mathbb{R}$, which you can intuitively understand to be time.
The coordinates $x(t)$ and $y(t)$ are functions of the parameter $t$ given in your case by $x(t) =\sin(t)$ and $y(t)=\sin(4t)$. The figure $$\{(\sin(t),\sin(4t)\mid t\in\mathbb{R}\}$$
is actually called a Lissajous curve. The direction and speed at which the figure is "drawn" depends on how you go trough the parameter $t$. If you start at $t=0$, you start at the origin $(0,0)$ and as $t$ increases, the point $(x(t),y(t))$ starts moving to the Northeast. However, you could just as well look at $$\{(\sin(-t),\sin(-4t)\mid t\in\mathbb{R}\}.$$
The curve in the $x$-$y$ plane will still be the same Lissajous curve, but if you start at $t=0$ and let $t$ increase and follow $(x(t),y(t))$, the point will move to the Southwest instead. In this case you're following the curve in the other direction.
So it all depends on $t$. In your original parametric equations, you are following the points by "walking" along the curve starting at the origin $(0,0)$ and moving to the Northeast. This video might help you as well to visualize this.