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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 1 - Trigonometric FunctionsProblema 34
Capitolo 2, Problema 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

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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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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.
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Quadratic Formula

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.
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Quadratic Formula

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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Complex Numbers In Polar Form
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