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/r
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
2:04 minutesProblem 50
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 , r/y
Verified step by step guidance1
Identify the quadrant given in the problem, which is Quadrant I. In this quadrant, both x and y coordinates are positive, so \(x > 0\) and \(y > 0\).
Recall the formula for \(r\), the distance from the origin to the point \((x, y)\): \(r = \sqrt{x^2 + y^2}\). Since \(x^2\) and \(y^2\) are always non-negative, \(r\) is always positive.
Analyze the ratio given: \(\frac{r}{y}\). Since \(r > 0\) and \(y > 0\) in Quadrant I, both numerator and denominator are positive.
Because both numerator and denominator are positive, the ratio \(\frac{r}{y}\) must be positive.
To confirm your understanding, sketch the coordinate plane, plot a point in Quadrant I, and visually verify that \(r\) and \(y\) are positive, reinforcing why the ratio is positive.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
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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Distance from Origin (r)
The distance r from the origin to a point (x, y) is given by r = √(x² + y²), which is always positive. This value represents the radius in polar coordinates and is crucial for understanding ratios involving r.
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Sign of Ratios Involving Coordinates
The sign of a ratio like r/y depends on the signs of numerator and denominator. Since r is always positive, the sign of r/y depends solely on y's sign, which varies by quadrant, helping determine if the ratio is positive or negative.
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