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## Homework Statement

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

## Homework Equations

ds

^{2}=r

^{2}dtheta

^{2}+dr

^{2}

ds= integral from beta to alpha sqrt[r

^{2}+ (dr/d theta)

^{2}]dtheta

## 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.