Given that $\cot \theta = -\frac{3}{4}$ and that $\tan \theta$ and $\cos \theta$ have opposite signs, without evaluating $\theta$, find the exact values of each of the following.

1. Determine the quadrant of $\theta$ and find $\cos \theta$.

Since $\cot \theta = -\frac{3}{4}$, we have $\tan \theta = -\frac{4}{3}$.
As $\tan \theta$ is negative and $\tan \theta$ and $\cos \theta$ have opposite signs, $\cos \theta$ must be positive.
This means $\theta$ lies in the 4th quadrant.
Using a right-angled triangle with opposite side $4$ and adjacent side $3$, the hypotenuse is $\sqrt{3^2 + 4^2} = 5$.
Therefore, $\cos \theta = \frac{3}{5}$.

2. Evaluate $\cos(-\theta)$.

Using the property of the cosine function, $\cos(-\theta) = \cos \theta$.
$\cos(-\theta) = \frac{3}{5}$

1. Determine the values of $\sin \theta$ and $\cos \theta$ based on the given conditions

Given $\cot \theta = -\frac{3}{4}$, we have $\tan \theta = -\frac{4}{3}$.
Since $\tan \theta$ is negative and $\cos \theta$ has the opposite sign, $\cos \theta$ must be positive. This places $\theta$ in the 4th quadrant.
In the 4th quadrant, $\sin \theta$ is negative.
Using a right-angled triangle with opposite side $4$ and adjacent side $3$, the hypotenuse is $\sqrt{3^2 + 4^2} = 5$.
Thus, $\sin \theta = -\frac{4}{5}$ and $\cos \theta = \frac{3}{5}$.

2. Calculate $\sin 2\theta$ using the double angle formula

$\sin 2\theta = 2\sin \theta \cos \theta$
$\sin 2\theta = 2\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)$
$\sin 2\theta = -\frac{24}{25}$