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.) IV , x/r

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Recall that the ratio given is \( \frac{x}{r} \), where \( r = \sqrt{x^2 + y^2} \) is the distance from the origin to the point \( (x, y) \). Since \( r \) is a square root of sums of squares, it is always positive.

Identify the quadrant of the point. The problem states the point is in Quadrant IV. In Quadrant IV, the \( x \)-coordinate is positive and the \( y \)-coordinate is negative.

Since \( x > 0 \) in Quadrant IV and \( r > 0 \) always, the ratio \( \frac{x}{r} \) is a positive number divided by a positive number.

Therefore, the ratio \( \frac{x}{r} \) must be positive in Quadrant IV.

To confirm, you can sketch the coordinate plane, plot a point in Quadrant IV, and visualize that \( x \) is positive and \( r \) is the hypotenuse (always positive), reinforcing the positivity of the ratio.

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주요 개념

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Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant IV, x is positive and y is negative. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios involving x and y.

Distance from Origin (r)

The distance r from the origin to a point (x, y) is given by r = √(x² + y²). Since squares are always non-negative, r is always positive. This ensures that ratios involving r in the denominator maintain the sign of the numerator.

Sign of Ratios in Trigonometry

Ratios like x/r correspond to trigonometric functions (e.g., cosine θ). The sign of such ratios depends on the signs of numerator and denominator. Since r is positive, the sign of x/r depends solely on x, which is positive in Quadrant IV, making x/r positive.