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

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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 ($$ \sin(	heta) = \frac{y}{r} $$)**
   - **Sign:** **Negative**
   - **Reason:** $$ y $$ is negative and $$ r $$ is positive.

2. **Cosine Ratio ($$ \cos(	heta) = \frac{x}{r} $$)**
   - **Sign:** **Negative**
   - **Reason:** $$ x $$ is negative and $$ r $$ is positive.

3. **Tangent Ratio ($$ 	an(	heta) = \frac{y}{x} $$)**
   - **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:** $$ \frac{y}{r}, \frac{x}{r}, \sin(	heta), \cos(	heta) $$
- **Positive Ratios:** $$ \frac{y}{x}, \frac{x}{y}, 	an(	heta) $$

**Example:**
If you're asked whether $$ 	an(	heta) $$ 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!
In Quadrant III, the ratio $$ \frac{y}{x} $$ is **Positive**.

Suppose that the point is in the indicated quadrant. Decide whether the given ratio is positive or negative. Recall that .
III,
Choose whether the given ratio is positive or negative.
Nogative
Positive

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Suppose that the point is in the indicated quadrant. Decide whether the given ratio is positive or negative. Recall that . III, Choose whether the given ratio is positive or negative. Nogative Positive