If `7sin^2 theta + 3 cos^2 theta = 4`, then the value of `cos theta(0^@le theta le 90^@)` is

Options

A. `sqrt6/2`
B. `sqrt3/2`
C. `sqrt2/2`
D. `sqrt5/2`

Answer

To solve the equation \(7 \sin^2 \theta + 3 \cos^2 \theta = 4\) and find the value of \(\cos \theta\) for \(0^\circ \leq \theta \leq 90^\circ\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \]
We know that \(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, we can express \(\cos^2 \theta\) in terms of \(\sin^2 \theta\):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

### Step 2: Substitute \(\cos^2 \theta\)
Substituting \(\cos^2 \theta\) into the original equation gives:
\[ 7 \sin^2 \theta + 3(1 - \sin^2 \theta) = 4 \]
Expanding this, we get:
\[ 7 \sin^2 \theta + 3 - 3 \sin^2 \theta = 4 \]

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

### Step 4: Isolate \(\sin^2 \theta\)
Next, we isolate \(\sin^2 \theta\):
\[ 4 \sin^2 \theta = 4 - 3 \]
\[ 4 \sin^2 \theta = 1 \]
Dividing both sides by 4 gives:
\[ \sin^2 \theta = \frac{1}{4} \]

### Step 5: Find \(\sin \theta\)
Taking the square root of both sides, we find:
\[ \sin \theta = \frac{1}{2} \]
Since \(0^\circ \leq \theta \leq 90^\circ\), we have:
\[ \theta = 30^\circ \]

### Step 6: Find \(\cos \theta\)
Now, we can find \(\cos \theta\):
\[ \cos \theta = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Conclusion
Thus, the value of \(\cos \theta\) is:
\[ \cos \theta = \frac{\sqrt{3}}{2} \]

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