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Chapter 29: Chapter 29

Problem 924

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Find all angles on \(\left[0^{\circ}, 360^{\circ}\right)\) which satisfy $\sin 2 \mathrm{x}-\sqrt{2}\( \)\sin \mathrm{x}=0$

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The angles that satisfy the equation \(\sin 2x - \sqrt{2} \sin x = 0\) in the domain \([0^{\circ}, 360^{\circ})\) are \(x = 0^{\circ}\), \(45^{\circ}\), \(180^{\circ}\), and \(315^{\circ}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Rearrange the Equation

Rearrange the given equation to simplify it. Use the double-angle identity: \(\sin 2x = 2 \sin x \cos x \). So, we can write the original equation as: \(2 \sin x \cos x - \sqrt{2} \sin x =0\).

Step 2: Factor the Equation

Next, factor out a \(\sin x\) to isolate \(x\). This gives: \(\sin x(2 \cos x - \sqrt{2}) = 0 \).

Step 3: Solve for Trigonometric Equations

To solve for \(x\), set each factor equal to zero. This gives us two equations: 1) \(\sin x = 0 \), 2) \(2 \cos x = \sqrt{2} \) or \(\cos x = \sqrt{2}/2 = \frac{\sqrt{2}}{2} \) using the property of symmetry of cosine function.

Step 4: Solve First Equation \(\sin x = 0\)

The solutions for \(\sin x = 0\) in the domain \([0^{\circ}, 360^{\circ})\) are \(x = 0^{\circ}\) and \(x = 180^{\circ}\).

Step 5: Solve Second Equation \(\cos x = \frac{\sqrt{2}}{2}\)

The solutions for \(\cos x = \frac{\sqrt{2}}{2}\) in the domain \([0^{\circ}, 360^{\circ})\) are \(x = 45^{\circ}\) and \(x = 315^{\circ}\).

Step 6: Combine all Solutions

Combine all of the solutions: \(x = 0^{\circ}\), \(45^{\circ}\), \(180^{\circ}\), and \(315^{\circ}\) which satisfy the given equation \(\sin 2x - \sqrt{2} \sin x=0\) within the domain \([0^{\circ}, 360^{\circ})\).

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