Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , y/r

Verified step by step guidance

1. Recall that the ratio given is \( \frac{y}{r} \), where \( r = \sqrt{x^2 + y^2} \). Since \( r \) is the distance from the origin to the point \( (x, y) \), it is always positive.
2. Identify the quadrant: The point \( (x, y) \) is in Quadrant III. In this quadrant, both \( x \) and \( y \) coordinates are negative.
3. Since \( y \) is negative in Quadrant III and \( r \) is positive, the ratio \( \frac{y}{r} \) will have the sign of \( y \), which is negative.
4. Therefore, the ratio \( \frac{y}{r} \) is negative in Quadrant III.
5. To visualize this, sketch the coordinate plane, mark Quadrant III, plot a point with negative \( x \) and \( y \), and note that \( r \) is the hypotenuse (always positive), confirming the sign of the ratio.

Key Concepts

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific sign conventions for x and y coordinates. In Quadrant III, both x and y values are negative. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios involving x, y, and r.

Definition of r in the Coordinate Plane

The variable r represents the distance from the origin to the point (x, y), calculated as r = √(x² + y²). Since r is a distance, it is always positive regardless of the quadrant. This positivity affects the sign of ratios involving r.

Sign of Ratios Involving Coordinates and r

Ratios like y/r depend on the signs of numerator and denominator. Since r is always positive, the sign of y/r is determined solely by y. In Quadrant III, y is negative, so y/r is negative. This concept is key to evaluating the sign of trigonometric ratios.