Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 32

Chapter 2, Problem 32

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

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.

Identify the quadrant: The point \( (x, y) \) is in Quadrant III. In this quadrant, both \( x \) and \( y \) coordinates are negative.

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.

Therefore, the ratio \( \frac{y}{r} \) is negative in Quadrant III.

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.

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Key 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 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.

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