Solve : `7 sin^(2) theta + 3 cos^(2) theta = 4`.

Answer

To solve the equation \( 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
We can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

### Step 2: Substitute in the Original Equation
Substituting \( \cos^2 \theta \) in the original equation:
\[ 7 \sin^2 \theta + 3(1 - \sin^2 \theta) = 4 \]
This simplifies to:
\[ 7 \sin^2 \theta + 3 - 3 \sin^2 \theta = 4 \]

### Step 3: Combine Like Terms
Combine the terms involving \( \sin^2 \theta \):
\[ (7 - 3) \sin^2 \theta + 3 = 4 \]
This simplifies to:
\[ 4 \sin^2 \theta + 3 = 4 \]

### Step 4: Isolate \( \sin^2 \theta \)
Now, isolate \( \sin^2 \theta \):
\[ 4 \sin^2 \theta = 4 - 3 \]
\[ 4 \sin^2 \theta = 1 \]
\[ \sin^2 \theta = \frac{1}{4} \]

### Step 5: Solve for \( \sin \theta \)
Taking the square root of both sides gives:
\[ \sin \theta = \pm \frac{1}{2} \]

### Step 6: Find the General Solutions
The angles for which \( \sin \theta = \frac{1}{2} \) are:
\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2n\pi \]
The angles for which \( \sin \theta = -\frac{1}{2} \) are:
\[ \theta = \frac{7\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{11\pi}{6} + 2n\pi \]

### Final Answer
Thus, the general solutions for \( \theta \) are:
\[ \theta = \frac{\pi}{6} + 2n\pi, \quad \frac{5\pi}{6} + 2n\pi, \quad \frac{7\pi}{6} + 2n\pi, \quad \frac{11\pi}{6} + 2n\pi \]
where \( n \) is any integer.