If `3costheta-sintheta=(1)/(sqrt(2)),(0^(@)ltthetalt90^(@))`, then the value of `3sintheta+2costheta`.

Options

A. a) `(1)/(sqrt(2))`
B. b) `sqrt(2)`
C. c) `(3)/(sqrt(2))`
D. d) `(5)/(sqrt(2))`

Answer

To solve the equation \(3 \cos \theta - \sin \theta = \frac{1}{\sqrt{2}}\) for \(3 \sin \theta + 2 \cos \theta\), we can follow these steps:

### Step 1: Write the given equation
We start with the equation:
\[ 3 \cos \theta - \sin \theta = \frac{1}{\sqrt{2}} \]

### Step 2: Let \(x = 3 \sin \theta + 2 \cos \theta\)
We define a new variable:
\[ x = 3 \sin \theta + 2 \cos \theta \]

### Step 3: Square both equations
Now, we will square both equations:
1. Squaring the first equation:
\[ (3 \cos \theta - \sin \theta)^2 = \left(\frac{1}{\sqrt{2}}\right)^2 \]
This expands to:
\[ 9 \cos^2 \theta - 6 \cos \theta \sin \theta + \sin^2 \theta = \frac{1}{2} \]

2. Squaring the second equation:
\[ (3 \sin \theta + 2 \cos \theta)^2 = x^2 \]
This expands to:
\[ 9 \sin^2 \theta + 12 \sin \theta \cos \theta + 4 \cos^2 \theta = x^2 \]

### Step 4: Combine the squared equations
Now we add the two squared equations:
\[ (9 \cos^2 \theta - 6 \cos \theta \sin \theta + \sin^2 \theta) + (9 \sin^2 \theta + 12 \sin \theta \cos \theta + 4 \cos^2 \theta) = \frac{1}{2} + x^2 \]

### Step 5: Simplify the left side
Combining like terms on the left side:
\[ (9 \cos^2 \theta + 4 \cos^2 \theta) + (9 \sin^2 \theta + \sin^2 \theta) + (-6 \cos \theta \sin \theta + 12 \sin \theta \cos \theta) = \frac{1}{2} + x^2 \]
This simplifies to:
\[ 13 (\cos^2 \theta + \sin^2 \theta) + 6 \sin \theta \cos \theta = \frac{1}{2} + x^2 \]

### Step 6: Use the Pythagorean identity
Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\):
\[ 13 \cdot 1 + 6 \sin \theta \cos \theta = \frac{1}{2} + x^2 \]
This simplifies to:
\[ 13 + 6 \sin \theta \cos \theta = \frac{1}{2} + x^2 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ x^2 = 13 + 6 \sin \theta \cos \theta - \frac{1}{2} \]
\[ x^2 = 12.5 + 6 \sin \theta \cos \theta \]

### Step 8: Solve for \(x\)
To find \(x\), we need to express \(6 \sin \theta \cos \theta\):
Using the identity \(2 \sin \theta \cos \theta = \sin(2\theta)\), we have:
\[ 6 \sin \theta \cos \theta = 3 \sin(2\theta) \]
Thus:
\[ x^2 = 12.5 + 3 \sin(2\theta) \]

### Step 9: Find the maximum value of \(x\)
The maximum value of \(\sin(2\theta)\) is 1, so:
\[ x^2 = 12.5 + 3 \cdot 1 = 15.5 \]
Taking the square root gives:
\[ x = \sqrt{15.5} \]
Calculating gives:
\[ x = \frac{5}{\sqrt{2}} \]

### Final Answer
Thus, the value of \(3 \sin \theta + 2 \cos \theta\) is:
\[ \boxed{\frac{5}{\sqrt{2}}} \]