Question

Let r = 1 + cos θ.
Find the horizontal and vertical tangent lines to the polar curve.

Answer 1

x= cos(θ) +\[\cos ^{2}(\theta)\]

Answer 2

Go on. So far so good.

Answer 3

y= \[\sin(\theta) +\cos(\theta)\sin(\theta)\]

Answer 4

2 for 2. Now the derivatives!

Answer 5

dx/dθ= -sinθ(1+2cos(θ))

Answer 6

dy/dθ= \[\cos(\theta) +\cos ^{2}(\theta) -\sin ^{2}(\theta)\]

Answer 7

Just off the top of my head, I'm thinking I may prefer these:

\(\dfrac{dx}{d\theta} = -\sin(\theta) - 2\sin(\theta)\cos(\theta) = -sin(\theta) - \sin(2\theta)\)

\(\dfrac{dy}{d\theta} = \cos(\theta) + \cos(2\theta)\)

Maybe not. Just keeping my eyes open and my head available for alternate methods.

Answer 8

i'll start with dx/dθ. i'm getting θ to be 0, 2pi, 2pi/3, and 4pi/3

Answer 9

Well, you can't just "start with". If both are zero in the same place, that is an entirely different problem.

Answer 10

i meant to say that i'm going to find the vertical tangent lines first which is when dx/dθ = 0

Answer 11

you can interrupt lol. but i'm not understanding your hint

Answer 12

I'm a little puzzled that in your search for \(\dfrac{dx}{d\theta} = 0\), you managed to miss \(\theta = \pi\), since that's the important one about which I am hinting.

Answer 13

oh i canceled pi thinking it couldn't be one because it's undefined. i forgot that vertical tangent lines include them