Ch. 1 - Trigonometric Functions

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

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 34

Chapter 2, Problem 34

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

Verified step by step guidance

Recall that the point (x, y) lies in Quadrant IV. In this quadrant, the x-coordinate is positive and the y-coordinate is negative.

The ratio given is \( \frac{x}{y} \). Since \( x > 0 \) and \( y < 0 \) in Quadrant IV, the numerator is positive and the denominator is negative.

A positive number divided by a negative number results in a negative value.

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

To visualize this, sketch the coordinate plane, mark Quadrant IV, and plot a point with positive x and negative y to see why the ratio is negative.

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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 IV, x is positive and y is negative. Understanding these sign rules is essential for determining the sign of ratios involving x and y.

Sign of Ratios in Different Quadrants

The sign of a ratio like x/y depends on the signs of numerator and denominator. Since x and y have known signs in each quadrant, the ratio's sign can be deduced by dividing their signs. For example, in Quadrant IV, x/y is positive divided by negative, resulting in a negative ratio.

Distance from Origin (r = √(x² + y²))

The distance r from the origin to the point (x, y) is always positive and is calculated using the Pythagorean theorem. While r itself is positive, it helps relate x and y to trigonometric functions and confirms the point's position relative to the origin.

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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.) III , y/r

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