To determine in which quadrant [tex]\(\theta\)[/tex] lies, given that [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\cos \theta > 0\)[/tex], let's analyze the properties of the trigonometric functions in each of the four quadrants:
1. First Quadrant (0 to 90 degrees or [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians):
- [tex]\(\sin \theta > 0\)[/tex] (sine is positive)
- [tex]\(\cos \theta > 0\)[/tex] (cosine is positive)
2. Second Quadrant (90 to 180 degrees or [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians):
- [tex]\(\sin \theta > 0\)[/tex] (sine is positive)
- [tex]\(\cos \theta < 0\)[/tex] (cosine is negative)
3. Third Quadrant (180 to 270 degrees or [tex]\( \pi \)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians):
- [tex]\(\sin \theta < 0\)[/tex] (sine is negative)
- [tex]\(\cos \theta < 0\)[/tex] (cosine is negative)
4. Fourth Quadrant (270 to 360 degrees or [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\( 2\pi \)[/tex] radians):
- [tex]\(\sin \theta < 0\)[/tex] (sine is negative)
- [tex]\(\cos \theta > 0\)[/tex] (cosine is positive)
We are given that both [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\cos \theta > 0\)[/tex]. Examining the signs of the sine and cosine functions in each quadrant:
- Both sine and cosine are positive only in the first quadrant.
Therefore, the terminal point determined by [tex]\(\theta\)[/tex] is in:
C. quadrant 1 .
