Answer
$\frac{\sqrt{3}}{3}$
Work Step by Step
Find $dy/d\theta$:
$\frac{dy}{d\theta}=\frac{d}{d\theta}(r\sin\theta)=\frac{d}{d\theta}(2\cos\theta\sin\theta)=\frac{d}{d\theta}(\sin2\theta)=2\cos2\theta$
Find $dx/d\theta$:
$\frac{dx}{d\theta}=\frac{d}{d\theta}(r\cos \theta)=\frac{d}{d\theta}(2\cos \theta\cos \theta)=\frac{d}{d\theta}(2\cos^2\theta)=\frac{d}{d\theta}(\cos 2\theta+1)=-2\sin2\theta+0=-2\sin2\theta$
Find $dy/dx$:
$\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{2\cos 2\theta}{-2\sin2\theta}=-\cot2\theta$
Find the slope of the tangent line to the curve at $\theta=\pi/3$:
$m=\frac{dy}{dx}|_{\theta=\pi/3}=-\cot(2\cdot \pi/3)=-\cot(2\pi/3)=\sqrt{3}/3$
Thus, the slope is $\frac{\sqrt{3}}{3}$.
