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

Ask by Turnbull Erickson. in South Africa
Mar 14,2025

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\( \beta \) lies in Quadrant II.

Solución

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**.

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Extra Insights

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.

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