r.s = 2 cos(t) + 2 sin(t).
Thus r and s are at right angles when r.s=0, so we are looking for the
smallest positive root of:
cos(t) + sin(t) = 0,
which is where you had got to.
Now to get an idea of where this root is your best approach is to sketch
some curves. The point we seek is where:
cos(t) = -sin(t)
if we sketch these it appears that the first positive root is near t=3pi/4,
substituting this into the equation confirms that this is in fact a solution.