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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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Dear Katie, Welcome to math.SE. since you are a new user, we wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say what your thoughts on the problem are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level. Further, it would be better if you could typeset your problem so that it is easy for people to read. Kindly look here… for more details on typesetting. – user17762 Nov 2 '12 at 4:16
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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