Suppose that the point \( (x, y) \) is in the indicated quadrant. Decide whether the given ratio is positive or negative. Recall that \( r=\sqrt{x^{2}+y^{2}} \). III, Choose whether the given ratio is positive or negative. Nogative Positive

In **Quadrant III**, both the \( x \) and \( y \) coordinates of a point \( (x, y) \) are **negative**, while the radius \( r = \sqrt{x^2 + y^2} \) is always **positive**. Depending on the specific ratio you're examining, here's how you can determine its sign: 1. **Sine Ratio (\( \sin(\theta) = \frac{y}{r} \))** - **Sign:** **Negative** - **Reason:** \( y \) is negative and \( r \) is positive. 2. **Cosine Ratio (\( \cos(\theta) = \frac{x}{r} \))** - **Sign:** **Negative** - **Reason:** \( x \) is negative and \( r \) is positive. 3. **Tangent Ratio (\( \tan(\theta) = \frac{y}{x} \))** - **Sign:** **Positive** - **Reason:** Both \( y \) and \( x \) are negative, and a negative divided by a negative yields a positive result. 4. **Other Ratios:** - **\( \frac{y}{x} \) or \( \frac{x}{y} \):** **Positive** - Both \( y \) and \( x \) are negative, so their ratio is positive. - **\( r \):** **Positive** - By definition, \( r \) is always positive. **Summary for Quadrant III:** - **Negative Ratios:** \( \frac{y}{r}, \frac{x}{r}, \sin(\theta), \cos(\theta) \) - **Positive Ratios:** \( \frac{y}{x}, \frac{x}{y}, \tan(\theta) \) **Example:** If you're asked whether \( \tan(\theta) \) is positive or negative in Quadrant III: - **Answer:** **Positive** This is because both \( y \) and \( x \) are negative, making their ratio \( \frac{y}{x} \) positive. If you have a specific ratio in mind, feel free to provide it, and I can help determine its sign in Quadrant III!