If `0lexle360^(@), tan xlt0 and cos x tanx gt0`, then which of the following is a possible value for x?

Options

A. `30^(@)`
B. `150^(@)`
C. `210^(@)`
D. `330^(@)`

Answer

To solve the problem, we need to analyze the conditions given:

1. **Given Conditions**:
- \( 0 < x < 360^\circ \)
- \( \tan x < 0 \)
- \( \cos x \tan x > 0 \)

2. **Understanding the Conditions**:
- The condition \( \tan x < 0 \) implies that \( x \) must be in a quadrant where the tangent function is negative. The tangent function is negative in the second quadrant (90° to 180°) and the fourth quadrant (270° to 360°).
- The condition \( \cos x \tan x > 0 \) can be rewritten as \( \cos x \cdot \tan x > 0 \). Since \( \tan x = \frac{\sin x}{\cos x} \), we can express this as \( \cos x \cdot \frac{\sin x}{\cos x} > 0 \), which simplifies to \( \sin x > 0 \).

3. **Analyzing Quadrants**:
- In the first quadrant (0° to 90°), both sine and cosine are positive, hence tangent is positive.
- In the second quadrant (90° to 180°), sine is positive and cosine is negative, hence tangent is negative.
- In the third quadrant (180° to 270°), both sine and cosine are negative, hence tangent is positive.
- In the fourth quadrant (270° to 360°), sine is negative and cosine is positive, hence tangent is negative.

4. **Conclusion**:
- We need a quadrant where \( \sin x > 0 \) and \( \tan x < 0 \). This only occurs in the **second quadrant** (90° to 180°).
- Therefore, \( x \) must be in the range \( 90^\circ < x < 180^\circ \).

5. **Identifying Possible Values**:
- Among the options provided, we need to find a value that lies in the range of 90° to 180°.
- If the options are, for example, 30°, 150°, 210°, and 300°, the only value that fits the criteria is **150°**.

Thus, the possible value for \( x \) is **150°**.