Skip to main content
Ch. 6 - Inverse Circular Functions and Trigonometric Equations
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. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.2.47
Capitolo 7, Problema 6.2.47

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
cos θ + 1 = 0

Guida verificata passo dopo passo
1
Start with the given equation: \(\cos \theta + 1 = 0\).
Isolate the cosine term by subtracting 1 from both sides: \(\cos \theta = -1\).
Recall that \(\cos \theta = -1\) at specific angles on the unit circle. Identify all angles \(\theta\) where the cosine value is exactly \(-1\) within one full rotation (0° to 360° or 0 to \(2\pi\) radians).
Express the general solution for \(\theta\) in degrees and radians. Since cosine is periodic with period \(360^\circ\) or \(2\pi\) radians, write the solutions as \(\theta = 180^\circ + 360^\circ k\) or \(\theta = \pi + 2\pi k\), where \(k\) is any integer.
For the least possible nonnegative angle measure, provide the principal solution(s) within the interval \([0, 360^\circ)\) or \([0, 2\pi)\), and mention the general solution to cover all possible angles.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

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

Solving Basic Trigonometric Equations

This involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. For example, solving cos θ + 1 = 0 means finding θ where cos θ = -1, which occurs at specific standard angles.
Video consigliato:
Percorso guidato
4:34
How to Solve Linear Trigonometric Equations

Angle Measurement and Conversion

Understanding the difference between radians and degrees is essential, as the problem requires solutions in both units. Knowing how to convert between radians and degrees (1 radian = 180/π degrees) helps in expressing answers correctly and rounding them as specified.
Video consigliato:
Percorso guidato
5:31
Reference Angles on the Unit Circle

General Solutions and Principal Values

Trigonometric equations often have infinitely many solutions due to periodicity. The general solution expresses all possible angles, while principal values are the smallest nonnegative angles within one full rotation (0 to 2π radians or 0° to 360°). Writing answers using least possible nonnegative measures ensures clarity.
Video consigliato:
Percorso guidato
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Pratica correlata
Domanda del libro di testo

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2 cos² x + cos x ― 1 = 0

25
views
Domanda del libro di testo

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 2π). Express solutions to four decimal places.

2 sin 2x ― x³ + 1 = 0

34
views
Domanda del libro di testo

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.


√2 sin 3x - 1 = 0

22
views
Domanda del libro di testo

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate

tan 2x + sec 2x = 3

39
views
Domanda del libro di testo

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.


3 csc² x/2 = 2 sec x

36
views
Domanda del libro di testo

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.


√2 cos 2θ = -1

35
views