To determine the quadrant in which the terminal point determined by the angle [tex]\(\theta\)[/tex] lies, given that [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\cos \theta > 0\)[/tex], let's analyze the conditions of the trigonometric functions in each quadrant.
1. First quadrant: Both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are positive.
2. Second quadrant: [tex]\(\sin \theta\)[/tex] is positive, but [tex]\(\cos \theta\)[/tex] is negative.
3. Third quadrant: Both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are negative.
4. Fourth quadrant: [tex]\(\sin \theta\)[/tex] is negative, but [tex]\(\cos \theta\)[/tex] is positive.
Given:
- [tex]\(\sin \theta > 0\)[/tex]
- [tex]\(\cos \theta > 0\)[/tex]
According to the conditions listed above:
- In the first quadrant ([tex]\(\theta\)[/tex] between [tex]\(0\)[/tex] and [tex]\(90\)[/tex] degrees), both sine and cosine values are positive.
Since both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are positive, [tex]\(\theta\)[/tex] must lie in the first quadrant.
Therefore, the terminal point determined by [tex]\(\theta\)[/tex] is in:
D. quadrant 1
