Let's analyze the given conditions step by step:
### Step 1: Understand Trigonometric Functions
We are given two conditions involving trigonometric functions on an angle [tex]\(\theta\)[/tex]:
1. [tex]\(\sin \theta < 0\)[/tex]
2. [tex]\(\tan \theta > 0\)[/tex]
### Step 2: Determine the Quadrants
A complete circle is divided into four quadrants as follows:
- 1st Quadrant (0° to 90°): [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\tan \theta > 0\)[/tex]
- 2nd Quadrant (90° to 180°): [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\tan \theta < 0\)[/tex]
- 3rd Quadrant (180° to 270°): [tex]\(\sin \theta < 0\)[/tex] and [tex]\(\tan \theta > 0\)[/tex]
- 4th Quadrant (270° to 360°): [tex]\(\sin \theta < 0\)[/tex] and [tex]\(\tan \theta < 0\)[/tex]
### Step 3: Apply the Sine Condition
The condition [tex]\(\sin \theta < 0\)[/tex] is satisfied in the 3rd and 4th quadrants:
- 3rd Quadrant (180° to 270°)
- 4th Quadrant (270° to 360°)
### Step 4: Apply the Tangent Condition
The condition [tex]\(\tan \theta > 0\)[/tex] is satisfied in the 1st and 3rd quadrants:
- 1st Quadrant (0° to 90°)
- 3rd Quadrant (180° to 270°)
### Step 5: Find the Intersection
We need to find where both conditions ([tex]\(\sin \theta < 0\)[/tex] and [tex]\(\tan \theta > 0\)[/tex]) are satisfied simultaneously.
This happens where both conditions overlap, which is in the 3rd quadrant (180° to 270°).
### Conclusion
Thus, the angle [tex]\(\theta\)[/tex] must be within the range of the 3rd quadrant. Therefore, the correct option is:
A. [tex]\(180^{\circ} < \theta < 270^{\circ}\)[/tex]
So, the answer is:
[tex]\[ \boxed{1} \][/tex]
