Solve: `4 cos ^(2) theta = 3 . (theta ^(@) le theta lt 360 ^(@)).`

Answer

To solve the equation \( 4 \cos^2 \theta = 3 \) for \( \theta \) in the range \( 0^\circ \leq \theta < 360^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ 4 \cos^2 \theta = 3 \]

### Step 2: Divide both sides by 4
To isolate \( \cos^2 \theta \), divide both sides of the equation by 4:
\[ \cos^2 \theta = \frac{3}{4} \]

### Step 3: Take the square root
Next, take the square root of both sides. Remember to consider both the positive and negative roots:
\[ \cos \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \]

### Step 4: Find the angles corresponding to \( \cos \theta = \frac{\sqrt{3}}{2} \)
The cosine function is positive in the first and fourth quadrants. The reference angle where \( \cos \theta = \frac{\sqrt{3}}{2} \) is:
\[ \theta = 30^\circ \quad \text{(or } \frac{\pi}{6} \text{ radians)} \]
Thus, the angles in the specified range are:
\[ \theta = 30^\circ \quad \text{and} \quad \theta = 360^\circ - 30^\circ = 330^\circ \]

### Step 5: Find the angles corresponding to \( \cos \theta = -\frac{\sqrt{3}}{2} \)
The cosine function is negative in the second and third quadrants. The reference angle where \( \cos \theta = -\frac{\sqrt{3}}{2} \) is:
\[ \theta = 180^\circ - 30^\circ = 150^\circ \quad \text{and} \quad \theta = 180^\circ + 30^\circ = 210^\circ \]

### Step 6: Compile all solutions
The solutions for \( \theta \) in the range \( 0^\circ \leq \theta < 360^\circ \) are:
\[ \theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ \]

### Final Answer
Thus, the complete solution set is:
\[ \theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ \]