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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 6.3.55
Capitolo 7, Problema 6.3.55

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

Guida verificata passo dopo passo
1
Rewrite the given equation to isolate the expression for clarity: \(2 \sin(2x) - x^3 + 1 = 0\).
Understand that this equation involves both a trigonometric function \(\sin(2x)\) and a polynomial term \(x^3\), making it transcendental and not solvable by standard algebraic methods.
Use a graphing calculator to plot the function \(f(x) = 2 \sin(2x) - x^3 + 1\) over the interval \([0, 2\pi)\) to visually identify where the graph crosses the x-axis (i.e., where \(f(x) = 0\)).
Zoom in on each x-intercept found in the graph and use the calculator's root-finding feature (such as the zero or root function) to approximate the solutions to four decimal places.
List all solutions found within the interval \([0, 2\pi)\), ensuring none are missed by checking the entire interval carefully.

Concetti chiave

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

Trigonometric Functions and Their Properties

Understanding the sine function, especially sin(2x), is crucial as it involves a double-angle identity. Recognizing how the sine function behaves over the interval [0, 2π) helps in identifying possible solution points and their periodic nature.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Graphical Solution of Equations

When algebraic methods fail, graphing both sides of the equation or the entire expression helps visualize where the function crosses the x-axis. Using a graphing calculator allows for approximating roots by identifying intersection points within the specified interval.
Video consigliato:
Percorso guidato
4:25
Introduction to Trig Equations

Numerical Approximation and Root Finding

Since exact algebraic solutions are not possible, numerical methods like the calculator’s root-finding feature or iterative approximation are used. Expressing solutions to four decimal places requires understanding how to interpret and refine these approximations accurately.
Video consigliato:
Percorso guidato
07:37
Complex Roots
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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

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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

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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.

cos θ + 1 = 0

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Domanda del libro di testo

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

sin² θ ― 2 sin θ + 3 = 0

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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.

sin (θ/2) = csc (θ/2)

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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 2x = √3

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