For what values of θ on the polar curve r=θ, with 0≤θ≤2π , are the tangent lines horizontal? vertical?

Given that , then

The slope of a tangent line in the polar coordinate is given by:



Thus, we have:



Part A:

For horizontal tangent lines, m = 0.

Thus, we have:



Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:

θ = 0

θ = 2.02875783811043

θ = 4.91318043943488

Part B:

For vertical tangent lines,

Thus, we have:



Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:

θ = 4.91718592528713