Answer
a) $\dfrac{1}{2}$
b) $0$
c) $\dfrac{2\sqrt 3-1}{\sqrt 3-2}$
Work Step by Step
Take the derivative of the given equations and isolate the variables.
$\dfrac{dy}{dt}=\cos \theta+2 \sin 2 \theta$
and $\dfrac{dx}{dt}=-\sin \theta+2 \cos 2 \theta$
Now, slope: $\dfrac{dy}{dx}=\dfrac{\cos \theta+2 \sin 2 \theta}{-\sin \theta+2 \cos 2 \theta}$
a) At $\theta =0$
Slope: $\dfrac{dy}{dx}=\dfrac{1}{2}$
b) At $\theta =\pi/2$
Slope: $\dfrac{dy}{dx}=0$
c) At $\theta =4\pi/3$
Slope: $\dfrac{dy}{dx}=\dfrac{\cos (4\pi/3)+2 \sin 2 (4\pi/3)}{-\sin (4\pi/3)+2 \cos 2 (4\pi/3)}=\dfrac{2\sqrt 3-1}{\sqrt 3-2}$
