Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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## Solving Trigonometric Equations ### Problem Statement: Solve for all degree solutions and θ if 0° ≤ θ < 360°. Do not use a calculator. (Enter your answers as a comma-separated list.) \[ \sqrt{2} \tan \theta + 2 \sin \theta \tan \theta = 0 \] ### Solution: #### (a) All Degree Solutions Let \( k \) be any integer. The solutions are given in terms of \( k \): 1. \( \theta = 180^\circ + 360^\circ k, 225^\circ + 360^\circ k, 315^\circ + 360^\circ k \) This option is selected as the correct solution. 2. \( \theta = 360^\circ k, 450^\circ + 360^\circ k, 45^\circ + 180^\circ k \) 3. \( \theta = 0^\circ + 180^\circ k, 450^\circ + 360^\circ k, 450^\circ + 180^\circ k \) 4. \( \theta = 360^\circ k, 225^\circ + 360^\circ k, 315^\circ + 360^\circ k \) 5. \( \theta = 0^\circ + 180^\circ k, 225^\circ + 360^\circ k, 315^\circ + 360^\circ k \) The selected solution is noted with a blue circle (indicating the correct answer) and a red X mark outside the selection box. #### (b) Principal Solutions (0° ≤ θ < 360°) For this specific interval, the solution is: \[ \theta = 180^\circ, 225^\circ, 315^\circ \] The answer is mentioned in a box and marked with a red X, indicating that there might be an error. ### Notes: - Ensure to verify the solution by solving the trigonometric equation manually to understand why the provided answers have been marked correct or incorrect. - The multiple-choice format helps in identifying the general solutions explicitly while the boxed values target a specific interval requirement for \( θ \).
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Transcribed Image Text:## Solving Trigonometric Equations ### Problem Statement: Solve for all degree solutions and θ if 0° ≤ θ < 360°. Do not use a calculator. (Enter your answers as a comma-separated list.) \[ \sqrt{2} \tan \theta + 2 \sin \theta \tan \theta = 0 \] ### Solution: #### (a) All Degree Solutions Let \( k \) be any integer. The solutions are given in terms of \( k \): 1. \( \theta = 180^\circ + 360^\circ k, 225^\circ + 360^\circ k, 315^\circ + 360^\circ k \) This option is selected as the correct solution. 2. \( \theta = 360^\circ k, 450^\circ + 360^\circ k, 45^\circ + 180^\circ k \) 3. \( \theta = 0^\circ + 180^\circ k, 450^\circ + 360^\circ k, 450^\circ + 180^\circ k \) 4. \( \theta = 360^\circ k, 225^\circ + 360^\circ k, 315^\circ + 360^\circ k \) 5. \( \theta = 0^\circ + 180^\circ k, 225^\circ + 360^\circ k, 315^\circ + 360^\circ k \) The selected solution is noted with a blue circle (indicating the correct answer) and a red X mark outside the selection box. #### (b) Principal Solutions (0° ≤ θ < 360°) For this specific interval, the solution is: \[ \theta = 180^\circ, 225^\circ, 315^\circ \] The answer is mentioned in a box and marked with a red X, indicating that there might be an error. ### Notes: - Ensure to verify the solution by solving the trigonometric equation manually to understand why the provided answers have been marked correct or incorrect. - The multiple-choice format helps in identifying the general solutions explicitly while the boxed values target a specific interval requirement for \( θ \).
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