Provide an Appropriate Response $r = \cos ( n \theta )$

Essay

Provide an appropriate response.
-Show that the arc length of one petal of the rose $r = \cos ( n \theta )$ is given by $2 \int _ { 0 } ^ { \pi / ( 2 n ) } \sqrt { 1 + \left( n ^ { 2 } - 1 \right) \sin ^ { 2 } ( n \theta ) } d \theta$ and use this formula to help make a conjecture about the limit of such $\operatorname { arc }$ lengths as $\mathrm { n } \rightarrow + \infty$ .