#### Answer

(a) The slope of the tangent line is $~~-tan~\theta$
(b) The tangent line is horizontal when $~~\theta = \pi~n~~~$ where $n$ is an integer
The tangent line is vertical when $~~\theta = \frac{\pi}{2}+\pi~n~~~$ where $n$ is an integer
(c) The tangent line has a slope of $1$ when $~~\theta = \frac{3\pi}{4}+\pi~n~~~$ where $n$ is an integer
The tangent line has a slope of $-1$ when $~~\theta = \frac{\pi}{4}+\pi~n~~~$ where $n$ is an integer

#### Work Step by Step

(a) We can find $\frac{dy}{dx}$:
$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{3asin^2~\theta~cos~\theta}{3a~cos^2~\theta~(-sin~\theta)} = -\frac{sin~\theta}{cos~\theta} = -tan~\theta$
Since the slope of the tangent line is $\frac{dy}{dx}$, the slope of the tangent line is $-tan~\theta$
(b) The slope is $0$ when the tangent line is horizontal. We can find $\theta$ when $\frac{dy}{dx} = 0$:
$-tan~\theta = 0$
$sin~\theta = 0$
$\theta = \pi~n~~~$ where $n$ is an integer
The slope is undefined when the tangent line is vertical. We can find $\theta$ when $\frac{dy}{dx}$ is undefined:
$-tan~\theta$ is undefined
$cos~\theta = 0$
$\theta = \frac{\pi}{2}+\pi~n~~~$ where $n$ is an integer
(c) We can find $\theta$ when $\frac{dy}{dx} = 1$:
$-tan~\theta = 1$
$tan~\theta = -1$
$sin~\theta = -cos~\theta$
$\theta = \frac{3\pi}{4}+\pi~n~~~$ where $n$ is an integer
We can find $\theta$ when $\frac{dy}{dx} = -1$:
$-tan~\theta = -1$
$tan~\theta = 1$
$sin~\theta = cos~\theta$
$\theta = \frac{\pi}{4}+\pi~n~~~$ where $n$ is an integer
The tangent line has a slope of $1$ when $\theta = \frac{3\pi}{4}+\pi~n~~~$ where $n$ is an integer
The tangent line has a slope of $-1$ when $\theta = \frac{\pi}{4}+\pi~n~~~$ where $n$ is an integer