If `theta` be an acute angle and `7 sin^(2) theta + 3 cos^(2) theta = 4 `, then the value of `tan theta` is

Options

A. `sqrt3`
B. `1/sqrt3`
C. `1`
D. `0`

Answer

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

### Step 1: Rewrite the equation using the Pythagorean identity
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). We can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting this into the original equation gives:
\[ 7 \sin^2 \theta + 3(1 - \sin^2 \theta) = 4 \]

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

### Step 4: Solve for \( \sin \theta \)
Taking the square root of both sides gives:
\[ \sin \theta = \frac{1}{2} \]
Since \( \theta \) is an acute angle, we take the positive root.

### Step 5: Find \( \cos \theta \)
Using the Pythagorean identity again:
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \]
Taking the square root gives:
\[ \cos \theta = \frac{\sqrt{3}}{2} \]

### Step 6: Calculate \( \tan \theta \)
The tangent function is defined as:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting the values we found:
\[ \tan \theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

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