Answer
\[-\sqrt{3}\]
Work Step by Step
\[r=\cos\left(\frac{\theta}{3}\right)\]
Differentiate $r$ with respect to $\theta$:
\[\frac{dr}{d\theta}=\frac{-1}{3}\sin\left(\frac{\theta}{3}\right)\]
Slope of tangent line to the polar curve is given by:
\[m=\frac{\frac{dr}{d\theta}\sin\theta+r\cos\theta}{\frac{dr}{d\theta}\cos\theta-r\sin\theta}\]
\[m=\frac{\frac{-1}{3}\sin\left(\frac{\theta}{3}\right)\sin\theta+\cos\left(\frac{\theta}{3}\right)\cos\theta}{\frac{-1}{3}\sin\left(\frac{\theta}{3}\right)\cos\theta-\cos\left(\frac{\theta}{3}\right)\sin\theta}\]
at $\theta=\pi$
\[m=\displaystyle\frac{\frac{-1}{3}\sin\left(\frac{\pi}{3}\right)\sin\pi+\cos\left(\frac{\pi}{3}\right)\cos\pi}{\frac{-1}{3}\sin\left(\frac{\pi}{3}\right)\cos\pi-\cos\left(\frac{\pi}{3}\right)\sin\pi}\]
\[\Rightarrow m=\frac{0+\frac{1}{2}(-1)}{-\frac{1}{3}\left(\frac{\sqrt{3}}{2}\right)(-1)}\]
\[\Rightarrow m=\frac{\frac{-1}{2}}{\frac{1}{2\sqrt{3}}}\]
\[m=-\sqrt{3}\]
