If `5sin^(2)theta+3cos^(2)theta=4`. Find the value of `sin theta and cos theta`:

Options

A. `pm(1)/(sqrt2), pm(1)/(sqrt2)`
B. `pm(sqrt3)/(2),pmsqrt2`
C. `(sqrt3)/(2),(1)/(sqrt2)`
D. none of these

Answer

To solve the equation \(5\sin^2\theta + 3\cos^2\theta = 4\) and find the values of \(\sin\theta\) and \(\cos\theta\), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sin^2\theta + \cos^2\theta = 1 \]
From this, we can express \(\cos^2\theta\) in terms of \(\sin^2\theta\):
\[ \cos^2\theta = 1 - \sin^2\theta \]

### Step 2: Substitute \(\cos^2\theta\) in the Original Equation
Substituting \(\cos^2\theta\) into the original equation gives:
\[ 5\sin^2\theta + 3(1 - \sin^2\theta) = 4 \]

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ 5\sin^2\theta + 3 - 3\sin^2\theta = 4 \]
Combine like terms:
\[ (5\sin^2\theta - 3\sin^2\theta) + 3 = 4 \]
This simplifies to:
\[ 2\sin^2\theta + 3 = 4 \]

### Step 4: Isolate \(\sin^2\theta\)
Now, isolate \(\sin^2\theta\):
\[ 2\sin^2\theta = 4 - 3 \]
\[ 2\sin^2\theta = 1 \]
\[ \sin^2\theta = \frac{1}{2} \]

### Step 5: Find \(\sin\theta\)
Taking the square root of both sides gives:
\[ \sin\theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### Step 6: Find \(\cos\theta\)
Now, we can find \(\cos^2\theta\) using the Pythagorean identity:
\[ \cos^2\theta = 1 - \sin^2\theta = 1 - \frac{1}{2} = \frac{1}{2} \]
Taking the square root gives:
\[ \cos\theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### Final Answer
Thus, the values of \(\sin\theta\) and \(\cos\theta\) are:
\[ \sin\theta = \pm \frac{\sqrt{2}}{2}, \quad \cos\theta = \pm \frac{\sqrt{2}}{2} \]