If `7sin^(2)theta+3cos^(2)theta=4`, then find `tan theta`.

Options

A. `(1)/(sqrt(3))`
B. `(2)/(sqrt(3))`
C. `sqrt(3)`
D. `1`

Explanation

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

### Step 1: Rewrite the equation
We start with the equation:
\[ 7\sin^2\theta + 3\cos^2\theta = 4 \]

### Step 2: Use the Pythagorean identity
Recall 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 3: Substitute \( \cos^2\theta \) into the equation
Substituting \( \cos^2\theta \) into the original equation gives:
\[ 7\sin^2\theta + 3(1 - \sin^2\theta) = 4 \]

### Step 4: Simplify the equation
Expanding the equation:
\[ 7\sin^2\theta + 3 - 3\sin^2\theta = 4 \]
Combine like terms:
\[ (7 - 3)\sin^2\theta + 3 = 4 \]
This simplifies to:
\[ 4\sin^2\theta + 3 = 4 \]

### Step 5: Isolate \( \sin^2\theta \)
Subtract 3 from both sides:
\[ 4\sin^2\theta = 1 \]
Now, divide by 4:
\[ \sin^2\theta = \frac{1}{4} \]

### Step 6: Solve for \( \sin\theta \)
Taking the square root of both sides gives:
\[ \sin\theta = \frac{1}{2} \]

### Step 7: Find \( \theta \)
The angle \( \theta \) for which \( \sin\theta = \frac{1}{2} \) is:
\[ \theta = 30^\circ \]

### Step 8: Find \( \tan\theta \)
Now, we can find \( \tan\theta \):
\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \]
Since \( \sin 30^\circ = \frac{1}{2} \) and \( \cos 30^\circ = \frac{\sqrt{3}}{2} \):
\[ \tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \tan\theta \) is:
\[ \tan\theta = \frac{1}{\sqrt{3}} \]

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