Ch. 1 - Trigonometric Functions

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

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 40

Chapter 2, Problem 40

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.) II , y/x

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Recall that the ratio given is \( \frac{y}{x} \), where \(x\) and \(y\) are the coordinates of a point in the plane.

Identify the signs of \(x\) and \(y\) in Quadrant II. In this quadrant, \(x < 0\) (negative) and \(y > 0\) (positive).

Since \(y\) is positive and \(x\) is negative, the ratio \( \frac{y}{x} \) is a positive number divided by a negative number.

Dividing a positive number by a negative number results in a negative value, so \( \frac{y}{x} < 0 \) in Quadrant II.

Therefore, the ratio \( \frac{y}{x} \) is negative when the point \((x, y)\) lies in Quadrant II.

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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 signs for x and y coordinates. In Quadrant II, x is negative and y is positive. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios like y/x.

Sign of Ratios in Different Quadrants

The sign of a ratio such as y/x depends on the signs of y and x individually. Since y is positive and x is negative in Quadrant II, the ratio y/x will be negative. This concept is crucial for evaluating trigonometric ratios based on point location.

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 is not directly needed to find the sign of y/x, it is fundamental in defining trigonometric functions and understanding the point's position.

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Related Practice

Textbook Question

Perform each calculation.

110° 25' + 32° 55'

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Textbook Question

Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. cos θ > 0 , sec θ > 0

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Textbook Question

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.

cos θ = ―5/8 , and θ is in quadrant III

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Textbook Question

Perform each calculation. See Example 3. 75° 15' + 83° 32'

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Textbook Question

Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

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Textbook Question

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.

sec θ = ―√5 , and θ is in quadrant II

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