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 \( \beta \) lies based on the given conditions, we first analyze \( \sin \beta > 0 \). This indicates that \( \beta \) must be in either the first or second quadrant since sine is positive in those quadrants. Next, we look at the condition \( \tan \beta < 0 \). The tangent function is positive in the first quadrant and negative in the second quadrant. Therefore, combining both conditions, we conclude that \( \beta \) must be in the second quadrant, where sine is positive and tangent is negative. So, \( \beta \) lies in the second quadrant.