In working with polar coordinates, there will is a need for leveraging trigonometric identities, sometimes in creative ways. In this problem, we will walk through one such example.

Find the length of the curve $r=1-sin()$ on the interval $02$

We start by considering the graph of the function $r=f()=1-sin().$ We would expect this graph to be a cardioid with symmetry about the vertical line $=\frac{}{2}.$

$1-sin()$

We start by applying the to find the length of the curve:

${\displaystyle \underset{}{\overset{}{}}}\sqrt[\phantom{2}]{r^2+(\frac{dr}{d}{)}^2}d$

What are the appropriate values to substitute into our integral? Type $"$pi$"$ or $"$Pi$"$ for $$ and "theta" for $.$ Assume that we are not using symmetry to to set$-$up this initial integral.