For the angle $$ heta $$ in standard position, determine the quadrant where the terminal side of $$ heta $$ lies. $$ \csc heta0 $$ and $$ an heta

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For the angle $$ 	heta $$ in standard position, determine the quadrant where the terminal side of $$ 	heta $$ lies. $$ \csc 	heta>0 $$ and $$ 	an 	heta<0 $$ Quadrant:

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To determine the quadrant in which the terminal side of $$ heta $$ lies, given that $$ \csc heta > 0 $$ and $$ an heta < 0 $$, let's analyze each condition: 1. **$$ \csc heta > 0 $$** - $$ \csc heta = \frac{1}{\sin heta} $$ - Therefore, $$ \csc heta > 0 $$ implies $$ \sin heta > 0 $$. - The sine function is positive in **Quadrants I and II**. 2. **$$ an heta < 0 $$** - $$ an heta = \frac{\sin heta}{\cos heta} $$ - $$ an heta < 0 $$ implies that $$ \sin heta $$ and $$ \cos heta $$ have **opposite signs**. - This occurs in **Quadrants II and IV**. 3. **Combining both conditions:** - From $$ \sin heta > 0 $$, $$ heta $$ must be in Quadrant I or II. - From $$ an heta < 0 $$, $$ heta $$ must be in Quadrant II or IV. - The only quadrant that satisfies both conditions is **Quadrant II**. **Answer:** Quadrant II Quadrant II

For the angle \( \theta \) in standard position, determine the quadrant where the terminal side of \( \theta \) lies. \( \csc \theta>0 \) and \( \tan \theta<0 \) Quadrant:

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