Intersection of parametric function with coordinate axes

Question

Me and a friend could need some assistance:

$$x(t)=t\cos t$$ $$y(t)=t\sin t$$

We've been given the assignment to find where our parametric function intersects the x- and y-axes. But we can't really seem to find any solutions in our CAT programs.

We've got a solution by defining $y(t)=0$ and solving for $t$ which only gives us 0…

Answer 1

HINT:

• $\sin t=0$ if $t=0,\pm\pi,\pm2\pi,\dots$
• $\cos t=0$ if $t=\pm\frac12\pi,\pm\frac32\pi,\pm\frac52\pi,\dots$

Can you take it from here?

Answer 2

Intersection with the $x$-axis means $y = 0$: $$ x = t \cos t \\ 0 = t \sin t $$ The easiest solution is at $t = 0$. If $t \ne 0$ you can use $$ 0 = \sin(t) \Rightarrow t \in \{ \pm \pi, \pm 2 \pi, \pm 3 \pi \} $$ (because you can divide the second equation by $t$ on both sides) to have $t \in [-10, 10]$.

Intersection with the $y$-axis means $x = 0$: $$ 0 = t \cos t \\ y = t \sin t $$ Again $t = 0$ is a solution. For $t \ne 0$ we use $$ 0 = \cos t \Rightarrow t \in \{ \pm\pi/2, \pm 3 \pi /2, \pm 5 \pi /2 \} $$

Answer 3

the equation $$t\sin(t)=0$$ gives us $$t=0$$ or $$t=k\pi$$ with $$k \in \mathbb{Z}$$

Answer 4

To obtain the other solutions with Maple, use the allsolutions option to the solve command: solve(t*sin(t) = 0, allsolutions).

Answer 5

You may find the $y$ and $x$ intersections as follows:

[> restart:
   inty := {solve([t*sin(t) = 0, -10 < t, t < 10], t, allsolutions, explicit)};
   intx := {solve([t*cos(t) = 0, -10 < t, t < 10], t, allsolutions, explicit)};



inty := {{t = 0}, {t = Pi}, {t = -3*Pi}, {t = -2*Pi}, {t = -Pi}, {t = 2*Pi}, {t = 3*Pi}}

intx := {{t = 0}, {t = -(5/2)*Pi}, {t = -(3/2)*Pi}, {t = -(1/2)*Pi}, {t = (1/2)*Pi}, {t = (3/2)*Pi}, {t = (5/2)*Pi}}