We are given that $\sec x = -\frac{3}{2}$ and $\tan x > 0$, and we are tasked with finding the value of $\sin x$.

Step 1: Use the identity for secant

Recall that $\sec x = \frac{1}{\cos x}$, so: $\sec x = -\frac{3}{2} \implies \cos x = -\frac{2}{3}.$

Step 2: Use the Pythagorean identity

The Pythagorean identity states: $\sin^2 x + \cos^2 x = 1.$ Substituting $\cos x = -\frac{2}{3}$: $\sin^2 x + \left( -\frac{2}{3} \right)^2 = 1,$ $\sin^2 x + \frac{4}{9} = 1.$ Solve for $\sin^2 x$: $\sin^2 x = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9}.$ Thus: $\sin x = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3}.$

Step 3: Determine the correct sign of $\sin x$

We are given that $\tan x > 0$. Since $\tan x = \frac{\sin x}{\cos x}$, and $\cos x = -\frac{2}{3}$ (negative), $\sin x$ must also be negative for $\tan x$ to be positive. Therefore: $\sin x = -\frac{\sqrt{5}}{3}.$

Thus, the value of $\sin x$ is $-\frac{\sqrt{5}}{3}$.

Would you like further clarification on any steps?

Here are 5 related questions to expand your understanding:

1. How do you find the value of $\tan x$ given $\sec x$ and $\sin x$?
2. What are the basic trigonometric identities involving $\sin$, $\cos$, and $\tan$?
3. How can you determine the quadrant in which an angle lies based on the signs of trigonometric functions?
4. Why is it necessary to check the sign of $\tan x$ when solving for $\sin x$?
5. What is the significance of the Pythagorean identity in solving trigonometric equations?

Tip: Always check the sign of trigonometric functions by considering the quadrant in which the angle lies, using the given conditions.