Solve `2 sin ^(2) phi = sin phi, ( 0^(@) le phi lt 360^(@)).`

Answer

To solve the equation \( 2 \sin^2 \phi = \sin \phi \) for \( 0^\circ \leq \phi < 360^\circ \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ 2 \sin^2 \phi = \sin \phi \]
We can rearrange it to form a standard quadratic equation:
\[ 2 \sin^2 \phi - \sin \phi = 0 \]

### Step 2: Factoring the Equation
Next, we can factor out \( \sin \phi \):
\[ \sin \phi (2 \sin \phi - 1) = 0 \]

### Step 3: Setting Each Factor to Zero
Now, we set each factor equal to zero:
1. \( \sin \phi = 0 \)
2. \( 2 \sin \phi - 1 = 0 \)

### Step 4: Solving the First Factor
For the first factor \( \sin \phi = 0 \):
\[ \phi = 0^\circ \quad \text{or} \quad \phi = 180^\circ \]

### Step 5: Solving the Second Factor
For the second factor \( 2 \sin \phi - 1 = 0 \):
\[ 2 \sin \phi = 1 \implies \sin \phi = \frac{1}{2} \]
The angles that satisfy \( \sin \phi = \frac{1}{2} \) are:
\[ \phi = 30^\circ \quad \text{or} \quad \phi = 150^\circ \]

### Step 6: Combining the Solutions
Now, we combine all the solutions we found:
\[ \phi = 0^\circ, 30^\circ, 150^\circ, 180^\circ \]

### Final Answer
Thus, the possible values of \( \phi \) are:
\[ \phi = 0^\circ, 30^\circ, 150^\circ, 180^\circ \]