(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the total length of the cardioid r=a(1-cos theta)

2. Relevant equations

ds^{2}=r^{2}dtheta^{2}+dr^{2}

ds= integral from beta to alpha sqrt[r^{2}+ (dr/d theta)^{2}]dtheta

3. The attempt at a solution

dr=a(sin theta)d theta

ds^{2}=a^{2}(1-cos theta)^{2}d theta^{2}+ a^{2}sin^{2}theta (d theta^{2}), which reduces to:

ds=2^{1/2}a(1-cos theta)^{1/2}(d theta)

I'm good up to this point. My book says that the above equation for ds simplifies to:

2a|sin1/2theta|d theta

I don't see how to arrive at this simplification. Can someone please explain? Many thanks.

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# Homework Help: Arc lengths of polar curves

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