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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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
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Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.3.43
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.3.43Capitolo 7, Problema 6.3.43
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.
1 - sin x = cos 2x
Guida verificata passo dopo passo1
Rewrite the given equation: \(1 - \sin x = \cos 2x\).
Recall the double-angle identity for cosine: \(\cos 2x = 1 - 2\sin^2 x\). Substitute this into the equation to get \(1 - \sin x = 1 - 2\sin^2 x\).
Simplify the equation by subtracting 1 from both sides: \(-\sin x = -2\sin^2 x\). Then multiply both sides by -1 to get \(\sin x = 2\sin^2 x\).
Rewrite the equation as \(2\sin^2 x - \sin x = 0\) and factor it: \(\sin x (2\sin x - 1) = 0\).
Set each factor equal to zero and solve for \(x\):
1) \(\sin x = 0\)
2) \(2\sin x - 1 = 0 \Rightarrow \sin x = \frac{1}{2}\).
Find all solutions for \(x\) in radians within the specified domain, then convert to degrees if needed, and express answers as the least possible nonnegative angles.
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Concetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For this problem, the double-angle identity for cosine, cos 2x = 1 - 2sin²x or cos 2x = 2cos²x - 1, is essential to rewrite and simplify the equation for easier solving.
Video consigliato:
Percorso guidato
5:32
Fundamental Trigonometric Identities
Solving Trigonometric Equations
Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given domain. This includes considering the periodic nature of sine and cosine functions and expressing solutions using general formulas that account for all possible angles.
Video consigliato:
Percorso guidato
4:34How to Solve Linear Trigonometric Equations
Angle Measurement and Conversion
Understanding angle measurements in radians and degrees is crucial, as the problem requires solutions in both units. Converting between radians and degrees and expressing answers within the least nonnegative angle measure ensures clarity and correctness in the final solutions.
Video consigliato:
Percorso guidato
5:31Reference Angles on the Unit Circle
Pratica correlata
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 3θ = -1
36
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.
sin θ cos θ ― sin θ = 0
31
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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 tan θ) / (3 - tan² θ ) = 1
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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)
29
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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
22
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