4.1. In which quadrant will \( \beta \) lie if \( 0^{\circ} \leq \beta \leq 360^{\circ} \), and \( \sin \beta>0 \) and \( \tan \beta<0 \)?

To determine the quadrant in which the angle \( \beta \) lies given the conditions \( \sin \beta > 0 \) and \( \tan \beta < 0 \), we can analyze the signs of the trigonometric functions in each quadrant. 1. **Quadrant I**: - \( \sin \beta > 0 \) - \( \tan \beta > 0 \) 2. **Quadrant II**: - \( \sin \beta > 0 \) - \( \tan \beta < 0 \) 3. **Quadrant III**: - \( \sin \beta < 0 \) - \( \tan \beta > 0 \) 4. **Quadrant IV**: - \( \sin \beta < 0 \) - \( \tan \beta < 0 \) From the analysis: - The condition \( \sin \beta > 0 \) indicates that \( \beta \) must be in either Quadrant I or Quadrant II. - The condition \( \tan \beta < 0 \) indicates that \( \beta \) must be in Quadrant II (since tangent is negative in Quadrant II). Thus, combining both conditions, we conclude that \( \beta \) lies in **Quadrant II**.