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Problem 924

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

Expert verified
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 $$315^{\circ}$$.
See the step by step solution

## 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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