The value of `theta` for which `sqrt3 cos theta-sin theta=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 follow these steps:

### Step 1: Rearranging the equation
We start with the equation:
\[ \sqrt{3} \cos \theta - \sin \theta = 1 \]
We can rearrange this to isolate the trigonometric terms:
\[ \sqrt{3} \cos \theta - 1 = \sin \theta \]

### Step 2: Multiply by 2
To simplify the equation, we can multiply both sides by 2:
\[ 2(\sqrt{3} \cos \theta - 1) = 2 \sin \theta \]
This gives us:
\[ 2\sqrt{3} \cos \theta - 2 = 2 \sin \theta \]

### Step 3: Rewrite in terms of known angles
Now, we can express \( 2\sqrt{3} \cos \theta \) and \( 2 \sin \theta \) in terms of cosine and sine of known angles. We know that:
\[ 2\sqrt{3} \cos \theta = 2 \cdot \frac{\sqrt{3}}{2} \cos \theta = 2 \cos 30^\circ \cos \theta \]
And:
\[ 2 \sin \theta = 2 \cdot \frac{1}{2} \sin \theta = 2 \sin 30^\circ \sin \theta \]

### Step 4: Using the cosine of sum identity
Now we can apply the cosine of sum identity:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
This gives us:
\[ 2\sqrt{3} \cos \theta - 2 \sin \theta = 0 \]
which can be rewritten as:
\[ \cos(30^\circ + \theta) = \frac{1}{2} \]

### Step 5: Solving for \( \theta \)
The equation \( \cos(30^\circ + \theta) = \frac{1}{2} \) implies:
\[ 30^\circ + \theta = 60^\circ \quad \text{or} \quad 30^\circ + \theta = 300^\circ \]
From the first equation:
\[ \theta = 60^\circ - 30^\circ = 30^\circ \]
From the second equation:
\[ \theta = 300^\circ - 30^\circ = 270^\circ \]

### Step 6: Converting to radians
The values of \( \theta \) in radians are:
\[ 30^\circ = \frac{\pi}{6} \quad \text{and} \quad 270^\circ = \frac{3\pi}{2} \]

### Final Answer
Thus, the value of \( \theta \) for which \( \sqrt{3} \cos \theta - \sin \theta = 1 \) is:
\[ \theta = \frac{\pi}{6} \quad \text{or} \quad \theta = \frac{3\pi}{2} \]