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.) I , y/r

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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 regardless of the quadrant.

Identify the quadrant: The point \( (x, y) \) is in Quadrant I, where both \( x \) and \( y \) coordinates are positive.

Since \( y \) is positive in Quadrant I and \( r \) is always positive, the ratio \( \frac{y}{r} \) is the quotient of two positive numbers.

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

To visualize this, sketch the coordinate plane, plot a point in Quadrant I, draw the radius \( r \) from the origin to the point, and observe that both \( y \) and \( r \) are positive, confirming the positivity of the ratio.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant I, both x and y are positive, which affects the sign of ratios involving these values.

Definition of r in Trigonometry

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

Sign of Trigonometric Ratios

Trigonometric ratios like y/r correspond to sine of the angle formed by the point and the x-axis. The sign of y/r depends on the sign of y and the always positive r, so in Quadrant I, y/r is positive.