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

Options

A. `(1)/(sqrt(2))`
B. `sqrt(2)`
C. `(3)/(sqrt(2))`
D. `(sqrt(2))/(3)`

Answer

To solve the equation \(2 \cos \theta - \sin \theta = \frac{1}{\sqrt{2}}\) for \(0^\circ < \theta < 90^\circ\) and find the value of \(2 \sin \theta + \cos \theta\), we can follow these steps:

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

### Step 2: Substitute known values
We know that for \(\theta = 45^\circ\):
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \text{and} \quad \sin 45^\circ = \frac{1}{\sqrt{2}} \]
Substituting these values into the equation:
\[ 2 \left(\frac{1}{\sqrt{2}}\right) - \left(\frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \]
This simplifies to:
\[ \frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
\[ \frac{2 - 1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
Thus, the equation holds true, confirming that \(\theta = 45^\circ\).

### Step 3: Calculate \(2 \sin \theta + \cos \theta\)
Now, we need to find \(2 \sin \theta + \cos \theta\) using \(\theta = 45^\circ\):
\[ 2 \sin 45^\circ + \cos 45^\circ = 2 \left(\frac{1}{\sqrt{2}}\right) + \frac{1}{\sqrt{2}} \]
This simplifies to:
\[ \frac{2}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2 + 1}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]

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

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