A particle in d 2 spirals out from the origin according to

A particle in d = 2 spirals out from the origin according to x(t) = t cos t, y(t) = t sin t. 0 lessthanorequalto t lessthanorequalto infinity. Find expressions for the distance from the origin (which is not the are-length) as a function of the parameter t.

Solution

Here, d=2 x(t)= tcost y(t) = t sint

d^2 = x(t)^2 y(t)^2
(2)^2= t^2 cos^2(t) t^2 sin^2(t)

4 = t^2 * ( sin^2(t) cos^2(t) )
4 = t^2 * 1
2 = t

Formally this means t= 2 or t= - 2, but because t>=0 and distance needs to be positive, only the positive root is valid:

So t = 2