Answer :
To determine in which quadrant the terminal point determined by [tex]\(\theta\)[/tex] lies, we need to carefully consider the conditions given:
1. [tex]\(\cos \theta < 0\)[/tex]:
- The cosine function is negative in quadrants II and III.
2. [tex]\(\cot \theta > 0\)[/tex]:
- Cotangent is the reciprocal of the tangent function, [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
- Cotangent is positive when both sine and cosine have the same sign:
- [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\cos \theta > 0\)[/tex], which occurs in quadrant I (but this conflicts with [tex]\(\cos \theta < 0\)[/tex]).
- [tex]\(\sin \theta < 0\)[/tex] and [tex]\(\cos \theta < 0\)[/tex], which occurs in quadrant III.
Given that [tex]\(\cos \theta < 0\)[/tex] forces us to look in quadrants II or III, and [tex]\(\cot \theta > 0\)[/tex] indicates that sine and cosine must share the same sign, the only suitable quadrant satisfying both conditions is quadrant III.
Therefore, the terminal point determined by [tex]\(\theta\)[/tex] is in:
C. quadrant 3
1. [tex]\(\cos \theta < 0\)[/tex]:
- The cosine function is negative in quadrants II and III.
2. [tex]\(\cot \theta > 0\)[/tex]:
- Cotangent is the reciprocal of the tangent function, [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
- Cotangent is positive when both sine and cosine have the same sign:
- [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\cos \theta > 0\)[/tex], which occurs in quadrant I (but this conflicts with [tex]\(\cos \theta < 0\)[/tex]).
- [tex]\(\sin \theta < 0\)[/tex] and [tex]\(\cos \theta < 0\)[/tex], which occurs in quadrant III.
Given that [tex]\(\cos \theta < 0\)[/tex] forces us to look in quadrants II or III, and [tex]\(\cot \theta > 0\)[/tex] indicates that sine and cosine must share the same sign, the only suitable quadrant satisfying both conditions is quadrant III.
Therefore, the terminal point determined by [tex]\(\theta\)[/tex] is in:
C. quadrant 3
