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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.

If sec x = -3/2 and tan x > 0, what is the value of sin x?

Learn how to solve a trigonometry problem where sec x = -3/2 and tan x > 0. This problem involves key trigonometric identities like secant, sine, and tangent. Suitable for high school students.

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