How can I find the period of this? I should have said for one cycle.

Question

How can I find the period of this?

<10cos⁡(4πe^(-t)),10sin⁡(4πe^(-t))>

I should have said for one cycle.

anonymous · Answer

$e^{-t}$ is periodic in $t$ with a period of $2\pi i $

dumbcow · Answer

for this function, i dont think the period is constant

dumbcow · Answer

well imagine a particle accelerating along a cyclical path .... each cycle would take less time

anonymous · Answer

or not, i am a bit slow with my terminology

dumbcow · Answer

i think you could get it by saying...
$$e^{-t} = \frac{n}{2}$$
where n is the nth cycle
notice this will give multiples of 2pi inside cos/sin functions

anonymous · Answer

you can always reduce the exponent of t modulo 2pi though

dumbcow · Answer

@Charollete , just read your updated post
which cycle? for 0<t<2pi

anonymous · Answer

Sorry yeah the question says this "over what time interval does the object complete one lap?"

dumbcow · Answer

$$e^{-t} = \frac{1}{2}$$
$$t = \ln 2$$

anonymous · Answer

so the 4pi doesn't affect it then?

dumbcow · Answer

well thats where the 1/2 comes from
t>0 so e^-t is getting smaller than 1

starting at cos(4pi) ...one completer cycle for cosine is 2pi
how much time to get cos(2pi)
thus e^-t must be 1/2

dumbcow · Answer

just verified with wolfram http://www.wolframalpha.com/input/?i=parametric+plot%2810*cos%284pi*e%5E%28-t%29%29%2C+10*sin%284pi*e%5E%28-t%29%29%29+for+t%3D0+to+log%282%29

anonymous · Answer

Thank you! that makes perfect sense!

dumbcow · Answer

yw