Solve ` 2 sin theta cos theta = cos theta. ( 0^(@) < theta lt 360^(@)).`

Answer

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

### Step 1: Rearrange the equation
We start with the equation:
\[ 2 \sin \theta \cos \theta = \cos \theta \]
We can rearrange this to:
\[ 2 \sin \theta \cos \theta - \cos \theta = 0 \]

### Step 2: Factor out the common term
Next, we factor out \( \cos \theta \):
\[ \cos \theta (2 \sin \theta - 1) = 0 \]

### Step 3: Set each factor to zero
Now, we set each factor equal to zero:
1. \( \cos \theta = 0 \)
2. \( 2 \sin \theta - 1 = 0 \)

### Step 4: Solve for \( \theta \) from \( \cos \theta = 0 \)
The solutions for \( \cos \theta = 0 \) in the interval \( 0^\circ < \theta < 360^\circ \) are:
\[ \theta = 90^\circ \quad \text{and} \quad \theta = 270^\circ \]

### Step 5: Solve for \( \theta \) from \( 2 \sin \theta - 1 = 0 \)
Now, we solve the second equation:
\[ 2 \sin \theta = 1 \implies \sin \theta = \frac{1}{2} \]
The solutions for \( \sin \theta = \frac{1}{2} \) in the interval \( 0^\circ < \theta < 360^\circ \) are:
\[ \theta = 30^\circ \quad \text{and} \quad \theta = 150^\circ \]

### Step 6: Combine all solutions
Combining all the solutions from both factors, we have:
\[ \theta = 30^\circ, \quad 90^\circ, \quad 150^\circ, \quad 270^\circ \]

### Final Answer
Thus, the complete solution set is:
\[ \theta = 30^\circ, 90^\circ, 150^\circ, 270^\circ \]

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