It seems like your message is incomplete. You started with the condition $\tan \theta = -\frac{3}{4}$ with $\theta$ in quadrant IV, but the remainder of the message has some numbers and fractions that appear out of context.

Could you clarify your question or provide the complete details? For example, if you're looking for the values of trigonometric functions like $\sin \theta$, $\cos \theta$, or need to solve for another expression involving $\theta$, please let me know.

Here are some potential questions for expanding on this topic if helpful:

1. How do you find other trigonometric functions given $\tan \theta$ and the quadrant?
2. How does the quadrant affect the signs of trigonometric functions?
3. What are the steps to convert from tangent to sine and cosine in a specific quadrant?
4. What trigonometric identities might be useful in solving for unknown functions?
5. Can you find the exact value of $\theta$ given $\tan \theta$ in a specific quadrant?

Tip: In trigonometry, remembering the signs of functions in each quadrant (ASTC: All Students Take Calculus) can help determine the correct values when converting between trigonometric functions.