For the every `theta,0ltthetale90`, find the values of the following angle
`2sin^2theta+4cos^2theta=3`

Options

A. `30^@`
B. `60^@`
C. `45^@`
D. `80^@`

Answer

To solve the equation \( 2\sin^2\theta + 4\cos^2\theta = 3 \) for \( \theta \) in the range \( 0 < \theta < 90 \), we can follow these steps:

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

### Step 2: Use the Pythagorean identity
Recall the Pythagorean identity:
\[ \sin^2\theta + \cos^2\theta = 1 \]
We can express \( \sin^2\theta \) in terms of \( \cos^2\theta \):
\[ \sin^2\theta = 1 - \cos^2\theta \]

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

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

### Step 5: Isolate the cosine term
Subtract \( 2 \) from both sides:
\[ 2\cos^2\theta = 1 \]
Now, divide by \( 2 \):
\[ \cos^2\theta = \frac{1}{2} \]

### Step 6: Solve for cosine
Taking the square root of both sides gives:
\[ \cos\theta = \pm\frac{1}{\sqrt{2}} \]
Since \( \theta \) is in the range \( 0 < \theta < 90 \), we only consider the positive value:
\[ \cos\theta = \frac{1}{\sqrt{2}} \]

### Step 7: Find the angle
The angle \( \theta \) for which \( \cos\theta = \frac{1}{\sqrt{2}} \) is:
\[ \theta = 45^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 45^\circ \]

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