The value of `theta` for which `sqrt3costheta+sintheta=1` is:

Options

A. 0
B. `pi/3`
C. `pi/6`
D. `pi/2`

Answer

To solve the equation \( \sqrt{3} \cos \theta + \sin \theta = 1 \), we will analyze the equation step by step.

### Step 1: Write down the equation
We start with the equation:
\[ \sqrt{3} \cos \theta + \sin \theta = 1 \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate the trigonometric terms:
\[ \sqrt{3} \cos \theta + \sin \theta - 1 = 0 \]

### Step 3: Check possible values of \( \theta \)
We will check the possible values of \( \theta \) from the given options.

#### Option 1: \( \theta = 0 \)
Substituting \( \theta = 0 \):
\[ \sqrt{3} \cos(0) + \sin(0) = \sqrt{3} \cdot 1 + 0 = \sqrt{3} \neq 1 \]
So, option 1 is rejected.

#### Option 2: \( \theta = \frac{\pi}{3} \)
Substituting \( \theta = \frac{\pi}{3} \):
\[ \sqrt{3} \cos\left(\frac{\pi}{3}\right) + \sin\left(\frac{\pi}{3}\right) = \sqrt{3} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3} \neq 1 \]
So, option 2 is rejected.

#### Option 3: \( \theta = \frac{\pi}{6} \)
Substituting \( \theta = \frac{\pi}{6} \):
\[ \sqrt{3} \cos\left(\frac{\pi}{6}\right) + \sin\left(\frac{\pi}{6}\right) = \sqrt{3} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{3}{2} + \frac{1}{2} = 2 \neq 1 \]
So, option 3 is rejected.

#### Option 4: \( \theta = \frac{\pi}{2} \)
Substituting \( \theta = \frac{\pi}{2} \):
\[ \sqrt{3} \cos\left(\frac{\pi}{2}\right) + \sin\left(\frac{\pi}{2}\right) = \sqrt{3} \cdot 0 + 1 = 1 \]
This satisfies the equation, so option 4 is accepted.

### Conclusion
The value of \( \theta \) for which \( \sqrt{3} \cos \theta + \sin \theta = 1 \) is:
\[ \theta = \frac{\pi}{2} \]

---