Answer
$-\pi$
Work Step by Step
Find $dy/d\theta$:
$\frac{dy}{d\theta}=\frac{d}{d\theta}(r\sin\theta)=\frac{d}{d\theta}(\frac{1}{\theta}\cdot \sin\theta)=\frac{d}{d\theta}(\frac{\sin \theta}{\theta})=\frac{\cos\theta\cdot \theta -\sin \theta\cdot 1}{\theta^2}=\frac{\theta\cos\theta-\sin\theta}{\theta^2}$
Find $dx/d\theta$:
$\frac{dx}{d\theta}=\frac{d}{d\theta}(r\cos \theta)=\frac{d}{d\theta}(\frac{1}{\theta}\cdot \theta)=\frac{d}{d\theta}(\frac{\cos \theta}{\theta})=\frac{-\sin \theta\cdot \theta-\cos \theta\cdot \theta}{\theta^2}=-\frac{\theta\sin \theta+\cos \theta}{\theta^2}$
Find $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{\frac{\theta\cos\theta-\sin\theta}{\theta^2}}{-\frac{\theta\sin \theta+\cos \theta}{\theta^2}}=-\frac{\theta\cos \theta-\sin\theta}{\theta\sin \theta+\cos \theta}$
Find the slope of the tangent line at $\theta=\pi$:
$m=\frac{dy}{dx}|_{\theta=\pi}=-\frac{\pi\cos \theta-\sin \theta}{\pi \sin \theta+\cos \theta}=-\frac{\pi(-1)-0}{\pi\cdot 0+(-1)}=-\pi$
Thus, the slope is $-\pi$.
