Question

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

Answer 1

here, like usual, we are trying to find points where \(\Large {dy\over dx}=0 \text{ or }\infty\)

Answer 2

yes

Answer 3

for horizontal tangent line it'd just be dy= 0 which i'm getting values for theta to be 0, 2pi, pi/2, and 3pi/s

Answer 4

using chain rule, we have:
\[
\Large{dy\over dx}=\frac{{dr\over d\theta}\sin\theta+r\cos\theta}{{dr\over d\theta}\cos\theta-r\sin\theta}\\
{dy\over dx}=\frac{(-\sin\theta)\sin\theta+(1+\cos\theta)\cos\theta}{(-\sin\theta)\cos\theta-(1+\cos\theta)\sin\theta}
\]
use this now, to find for what \(\theta\) is the derivative "0" or infinite

Answer 5

HORIZONTAL tangent:
\[-\sin^2\theta+\cos\theta+\cos^2\theta=0\\
(1-\cos^2\theta)+\cos\theta+\cos^2\theta=0\\
2\cos^2\theta+\cos\theta-1=0\]
solve this and find the possilbe "theta"

Answer 6

then you can get the equation of horizontal tangent.

Answer 7

similarly, for the vertical, set the denominator to zero ad solve

Answer 8

for dx i'm getting -sin(theta) -2cos(theta)sin(theta)

Answer 9

@electrokid

Answer 10

what do you mean by "for dx"?