# Cardioid given by the polar equation $r = 1 − \cos(\theta)$

Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$.

(a) Find the intersection points of the curve with the line $\theta = \pi/4$.

(b) Find the intersection points of the curve with the circle $r = 1/2$.

(c) Find the slope of the tangent line to the curve at the point $(3/2,−2\pi/3)$.

(d) Find the length of the part of the curve in the fourth quadrant.

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What have you tried? If this is a homework problem, please tag it as such. –  Matthew Conroy Nov 2 '12 at 4:15
I have question in part (d), I got the answer of the langth, but the value is negative for θ from -π/2 to 0. As we know, the lenght cannot be negative, so whats the problem in here? –  user47890 Nov 2 '12 at 5:12

For (a), it's worth noting that $\theta=\pi/4$, $\theta=-3\pi/4$, and $r=0$ each put a point $(r,\theta)$ on the "line" $\theta=\pi/4$.

For (b), you need only find $-\pi\leq\theta\leq\pi$ such that $\frac12=1-\cos\theta$.

For (c), do you know how to find the tangent line at a point on a polar curve?

For (d), do you know how to find arc lengths?

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