To determine the correct range for [tex]\(\theta\)[/tex] where both [tex]\(\sin \theta < 0\)[/tex] and [tex]\(\tan \theta < 0\)[/tex], we need to understand the behavior of these trigonometric functions in different quadrants.
1. Sine function ([tex]\(\sin \theta\)[/tex]):
- [tex]\(\sin \theta > 0\)[/tex] in the first and second quadrants.
- [tex]\(\sin \theta < 0\)[/tex] in the third and fourth quadrants.
2. Tangent function ([tex]\(\tan \theta\)[/tex]):
- [tex]\(\tan \theta > 0\)[/tex] in the first and third quadrants.
- [tex]\(\tan \theta < 0\)[/tex] in the second and fourth quadrants.
Now, let's analyze the conditions given:
- [tex]\(\sin \theta < 0\)[/tex] implies [tex]\(\theta\)[/tex] must be in the third or fourth quadrant.
- [tex]\(\tan \theta < 0\)[/tex] implies [tex]\(\theta\)[/tex] must be in the second or fourth quadrant.
The only quadrant where both conditions ([tex]\(\sin \theta < 0\)[/tex] and [tex]\(\tan \theta < 0\)[/tex]) are satisfied is in the fourth quadrant.
In the fourth quadrant, the range of [tex]\(\theta\)[/tex] is [tex]\(270^\circ < \theta < 360^\circ\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{270^\circ < \theta < 360^\circ} \][/tex]
This corresponds to option B:
[tex]\[ \text{B. } 270^\circ < \theta < 360^\circ \][/tex]
