Graph the polar function and its tangents that are horizontal or vertical for r=sqrt(theta), 0

We are asked to graph the polar function r=sqrt(theta), 0<= theta <= 2pi , along with its vertical and horizontal tangents.
We can graph by plotting points:
theta:          r:0          0pi/6     .7236pi/3    1.0233pi/2    1.2533etc... yielding:

We can find the horizontal and vertical tangents by using:
(dy)/(dx)=((dy)/(d theta))/((dx)/(d theta))=(1/(2sqrt(theta)) sin theta + sqrt(theta) cos theta)/(1/(2sqrt(theta))cos theta-sqrt(theta) sin theta)
The horizontal tangents occur when the numerator ((dy)/(d theta) ) is zero, while the vertical tangents occur when the denominator is zero.
Solving numerically we get the horizontal tangents when x ~~ +- .653
and the vertical tangents when x ~~ -1.83
http://mathworld.wolfram.com/PolarCoordinates.html