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

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. none of these

Answer

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

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

### Step 2: Factor out the common term
Notice that we can factor out a 2 from the left-hand side:
\[ 2(\sin^2\theta + 2\cos^2\theta) = 3 \]

### Step 3: Divide both sides by 2
Now, divide both sides of the equation by 2:
\[ \sin^2\theta + 2\cos^2\theta = \frac{3}{2} \]

### Step 4: Use the Pythagorean identity
We know 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:
\[ (1 - \cos^2\theta) + 2\cos^2\theta = \frac{3}{2} \]

### Step 5: Simplify the equation
Now simplify the equation:
\[ 1 - \cos^2\theta + 2\cos^2\theta = \frac{3}{2} \]
This simplifies to:
\[ 1 + \cos^2\theta = \frac{3}{2} \]

### Step 6: Isolate \(\cos^2\theta\)
Subtract 1 from both sides:
\[ \cos^2\theta = \frac{3}{2} - 1 = \frac{1}{2} \]

### Step 7: Solve for \(\cos\theta\)
Taking the square root of both sides gives:
\[ \cos\theta = \pm\frac{1}{\sqrt{2}} = \pm\frac{\sqrt{2}}{2} \]

### Step 8: Determine the angles
The angles for which \(\cos\theta = \frac{\sqrt{2}}{2}\) are:
\[ \theta = 45^\circ \quad \text{and} \quad \theta = 315^\circ \]
The angle for which \(\cos\theta = -\frac{\sqrt{2}}{2}\) are:
\[ \theta = 135^\circ \quad \text{and} \quad \theta = 225^\circ \]

### Step 9: Check the options
From the options given (30°, 60°, 45°, none of these), the only angle that matches is:
\[ \theta = 45^\circ \]

### Final Answer
Thus, the solution to the equation \(2\sin^2\theta + 4\cos^2\theta = 3\) is:
\[ \theta = 45^\circ \]

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