If [tex]\(\cos \theta \ \textless \ 0\)[/tex] and [tex]\(\cot \theta \ \textgreater \ 0\)[/tex], then the terminal point determined by [tex]\(\theta\)[/tex] is in:

A. Quadrant 1
B. Quadrant 4
C. Quadrant 3
D. Quadrant 2



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