Why is there this pattern in the polar curves cos[at] U sin[at]

[flyingpig]

Homework Statement

Polar plot the following

sin[t] U cos[t]
sin[2t] U cos[2t]
sin[3t] U cos[3t]
sin[4t] U cos[4t]
sin[5t] U cos[5t]

Notice that cos[t] and sin[t] are the same graph rotated 90 degrees only? Interesting! Just like the cartesian graph.

Now here is something more interesting

The number of loops follow as

1t U 1 loop
2t U 4 loops
3t U 3 loops
4t U 8 loops
5t U 5 loop

Notice how the odd number stays the same and the even doubles? Why is there this pattern?

 

[LCKurtz]

They actually have the same number of loops. When you use the odd multiple of t the graph is plotted twice. Try going from 0 to pi.

 

[flyingpig]

But you can't see it! So it is halved.

 

[LCKurtz]

But you can't see it! So it is halved.

Believe whatever you want to.