Given that $\sin \theta = -\frac{3}{2}$ is mathematically impossible (since the sine of an angle always lies between -1 and 1), I believe there may be a typographical error in the problem. Let's assume that the correct value of $\sin \theta$ is $-\frac{3}{5}$, which is a possible value within the allowed range.

Additionally, we know that $\theta$ is not in Quadrant III (QIII). Since $\sin \theta$ is negative and $\theta$ is not in QIII, $\theta$ must be in Quadrant IV.

1. Find $\cos \theta$

Using the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ Substitute $\sin \theta = -\frac{3}{5}$: $\left( -\frac{3}{5} \right)^2 + \cos^2 \theta = 1$ $\frac{9}{25} + \cos^2 \theta = 1$ $\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}$ $\cos \theta = \pm \frac{4}{5}$ Since $\theta$ is in Quadrant IV, $\cos \theta$ is positive: $\cos \theta = \frac{4}{5}$

2. Find $\tan \theta$

The tangent function is defined as: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ Substitute $\sin \theta = -\frac{3}{5}$ and $\cos \theta = \frac{4}{5}$: $\tan \theta = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}$

3. Find $\cot \theta$

The cotangent function is the reciprocal of the tangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3}$

4. Find $\sec \theta$

The secant function is the reciprocal of the cosine: $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4}$

5. Find $\csc \theta$

The cosecant function is the reciprocal of the sine: $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}$

Summary of Results:

$\sin \theta = -\frac{3}{5}$ $\cos \theta = \frac{4}{5}$ $\tan \theta = -\frac{3}{4}$ $\cot \theta = -\frac{4}{3}$ $\sec \theta = \frac{5}{4}$ $\csc \theta = -\frac{5}{3}$

Would you like more details on any of the steps, or do you have any questions?

Here are 5 additional related questions:

1. What are the signs of trigonometric functions in different quadrants?
2. How do you derive the Pythagorean identities?
3. What are the reciprocal trigonometric identities?
4. How can you verify the identity $\sin^2 \theta + \cos^2 \theta = 1$?
5. How do trigonometric ratios change as angles move between quadrants?

Tip: When solving trigonometric problems, always check the quadrant to determine the correct signs of the trigonometric ratios.