Skip to main content
Ch. 2 - Acute Angles and Right Triangles
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. 2 - Acute Angles and Right TrianglesProblema 68
Capitolo 3, Problema 68

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3 sin θ = - —— 2

Guida verificata passo dopo passo
1
Recognize that the equation is \( \sin \theta = -\frac{\sqrt{3}}{2} \). We need to find all angles \( \theta \) in the interval \( [0^\circ, 360^\circ) \) where the sine value equals \( -\frac{\sqrt{3}}{2} \).
Recall the reference angle where \( \sin \theta = \frac{\sqrt{3}}{2} \) is \( 60^\circ \) because \( \sin 60^\circ = \frac{\sqrt{3}}{2} \). Since our sine value is negative, \( \theta \) must be in the quadrants where sine is negative.
Identify the quadrants where sine is negative: the third and fourth quadrants. So, the angles we seek are related to the reference angle \( 60^\circ \) but located in these quadrants.
Calculate the angles in the third and fourth quadrants by adding the reference angle to 180° and subtracting it from 360°, respectively. That is, \( \theta = 180^\circ + 60^\circ \) and \( \theta = 360^\circ - 60^\circ \).
Write the general solutions for \( \theta \) in the interval \( [0^\circ, 360^\circ) \) as \( \theta = 240^\circ \) and \( \theta = 300^\circ \).

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:
3m

Concetti chiave

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

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Angles are measured in degrees or radians, and the sine of an angle corresponds to the y-coordinate of the point on the unit circle. Understanding the interval [0°, 360°) means considering one full rotation around the circle.
Video consigliato:
Percorso guidato
06:11
Introduction to the Unit Circle

Sine Function and Its Values

The sine function gives the ratio of the opposite side to the hypotenuse in a right triangle, or the y-coordinate on the unit circle. Knowing that sin θ = -√3/2 indicates the angle's sine value is negative, which occurs in the third and fourth quadrants where y-values are negative.
Video consigliato:
Percorso guidato
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°

Reference Angles and Quadrants

Reference angles are acute angles used to find sine values for angles in different quadrants. Since sine is negative in the third and fourth quadrants, you find the reference angle with positive sine √3/2 (which is 60°), then determine θ by adjusting for the correct quadrant within [0°, 360°).
Video consigliato:
Percorso guidato
5:31
Reference Angles on the Unit Circle
Pratica correlata
Domanda del libro di testo

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3 cot θ = - —— 3

663
views
Domanda del libro di testo

Concept Check Work each problem. Find the equation of the line that passes through the origin and makes a 30° angle with the x-axis.

708
views
Domanda del libro di testo

Find the exact value of the variables in each figure.

817
views
Domanda del libro di testo

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. 1 cos θ = - — 2

698
views
Domanda del libro di testo

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. sec θ = -√2

641
views
Domanda del libro di testo

Concept Check Work each problem. What angle does the line y = √3x make with the positive x-axis?

663
views