Skip to main content
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 37
Capitolo 2, Problema 37

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

Guida verificata passo dopo passo
1
Recall that the ratio given is \( \frac{x}{r} \), where \( r = \sqrt{x^2 + y^2} \) is the distance from the origin to the point \( (x, y) \). Since \( r \) is a square root of sums of squares, it is always positive.
Identify the quadrant of the point. The problem states the point is in Quadrant IV. In Quadrant IV, the \( x \)-coordinate is positive and the \( y \)-coordinate is negative.
Since \( x > 0 \) in Quadrant IV and \( r > 0 \) always, the ratio \( \frac{x}{r} \) is a positive number divided by a positive number.
Therefore, the ratio \( \frac{x}{r} \) must be positive in Quadrant IV.
To confirm, you can sketch the coordinate plane, plot a point in Quadrant IV, and visualize that \( x \) is positive and \( r \) is the hypotenuse (always positive), reinforcing the positivity of the ratio.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant IV, x is positive and y is negative. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios involving x and y.
Video consigliato:
Percorso guidato
6:36
Quadratic Formula

Distance from Origin (r)

The distance r from the origin to a point (x, y) is given by r = √(x² + y²). Since squares are always non-negative, r is always positive. This ensures that ratios involving r in the denominator maintain the sign of the numerator.
Video consigliato:
Percorso guidato
6:58
Convert Points from Rectangular to Polar

Sign of Ratios in Trigonometry

Ratios like x/r correspond to trigonometric functions (e.g., cosine θ). The sign of such ratios depends on the signs of numerator and denominator. Since r is positive, the sign of x/r depends solely on x, which is positive in Quadrant IV, making x/r positive.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo

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

773
views
Domanda del libro di testo

Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

685
views
Domanda del libro di testo

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

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

938
views
Domanda del libro di testo

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . 12x + 5y = 0 , x ≥ 0 .

493
views
Domanda del libro di testo

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.

―345°

742
views
Domanda del libro di testo

Find the measure of the smaller angle formed by the hands of a clock at the following times. 8:20

712
views