Textbook Question
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/y
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
2:10 minutesProblem 47
Textbook Question
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
Verified step by step guidance1
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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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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