If `7 cos^2 theta +3 sin^2 theta =4 and 0 lt theta lt pi//2`, what is the value of `tan theta` ?

Options

A. `sqrt 7`
B. `7/3`
C. 3
D. `sqrt3`

Answer

To solve the equation \( 7 \cos^2 \theta + 3 \sin^2 \theta = 4 \) and find the value of \( \tan \theta \) for \( 0 < \theta < \frac{\pi}{2} \), we can follow these steps:

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

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

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

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

### Step 5: Solve for \( \cos \theta \)
Taking the square root of both sides:
\[ \cos \theta = \frac{1}{2} \]
Since \( 0 < \theta < \frac{\pi}{2} \), we take the positive root:
\[ \cos \theta = \frac{1}{2} \]

### Step 6: Find \( \theta \)
The angle \( \theta \) that satisfies \( \cos \theta = \frac{1}{2} \) in the first quadrant is:
\[ \theta = \frac{\pi}{3} \]

### Step 7: Calculate \( \tan \theta \)
Now we find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Using the known values for \( \theta = \frac{\pi}{3} \):
\[ \tan \frac{\pi}{3} = \sqrt{3} \]

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