If `5sin^(2)theta+4cos^(2)theta=(9)/(2)andltthetalt(pi)/(2)` then `tantheta` is equal to

Options

A. 1
B. 0
C. `-1`
D. `(1)/(4)`

Answer

To solve the equation \( 5\sin^2\theta + 4\cos^2\theta = \frac{9}{2} \) for \( \tan\theta \), we can follow these steps:

### Step 1: Substitute \( \cos^2\theta \)
We know that \( \cos^2\theta = 1 - \sin^2\theta \). So, we can rewrite the equation:
\[ 5\sin^2\theta + 4(1 - \sin^2\theta) = \frac{9}{2} \]

### Step 2: Simplify the equation
Distributing the \( 4 \) in the equation gives:
\[ 5\sin^2\theta + 4 - 4\sin^2\theta = \frac{9}{2} \]
Now, combine like terms:
\[ (5\sin^2\theta - 4\sin^2\theta) + 4 = \frac{9}{2} \]
This simplifies to:
\[ \sin^2\theta + 4 = \frac{9}{2} \]

### Step 3: Isolate \( \sin^2\theta \)
Subtract \( 4 \) from both sides:
\[ \sin^2\theta = \frac{9}{2} - 4 \]
Convert \( 4 \) to a fraction:
\[ \sin^2\theta = \frac{9}{2} - \frac{8}{2} = \frac{1}{2} \]

### Step 4: Take the square root
Taking the square root of both sides gives:
\[ \sin\theta = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 5: Find \( \theta \)
Since \( \theta \) is in the range \( (0, \frac{\pi}{2}) \), we have:
\[ \theta = 45^\circ \quad \text{or} \quad \theta = \frac{\pi}{4} \]

### Step 6: Calculate \( \tan\theta \)
Now, we can find \( \tan\theta \):
\[ \tan\theta = \tan\left(\frac{\pi}{4}\right) = 1 \]

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

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