If cos `theta+sqrt3 sin theta=2`, then the value of `theta` is equal to

Answer

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

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \cos \theta + \sqrt{3} \sin \theta = 2 \]
We can rewrite this equation as:
\[ \cos \theta + \sqrt{3} \sin \theta - 2 = 0 \]

### Step 2: Analyzing the Range of the Left Side
We know that the maximum value of \( \cos \theta \) is 1 and the maximum value of \( \sqrt{3} \sin \theta \) is \( \sqrt{3} \) (approximately 1.732). Therefore, the maximum value of \( \cos \theta + \sqrt{3} \sin \theta \) is:
\[ 1 + \sqrt{3} \approx 1 + 1.732 = 2.732 \]
Since the left-hand side can reach a maximum of approximately 2.732, it is possible for it to equal 2.

### Step 3: Finding Specific Values
To find specific values for \( \theta \), we can express \( \cos \theta \) and \( \sin \theta \) in terms of angles. We can rewrite the equation in a different form:
\[ \frac{1}{2} \cos \theta + \frac{\sqrt{3}}{2} \sin \theta = 1 \]
Here, we recognize that \( \frac{1}{2} \) corresponds to \( \cos 60^\circ \) and \( \frac{\sqrt{3}}{2} \) corresponds to \( \sin 60^\circ \).

### Step 4: Using Angle Addition Formula
Using the angle addition formula for cosine, we can express the left side as:
\[ \cos \theta \cos 60^\circ + \sin \theta \sin 60^\circ = \cos(\theta - 60^\circ) \]
Thus, we have:
\[ \cos(\theta - 60^\circ) = 1 \]

### Step 5: Solving for \( \theta \)
The equation \( \cos(\theta - 60^\circ) = 1 \) implies that:
\[ \theta - 60^\circ = 0 \quad \text{or} \quad \theta - 60^\circ = 360^\circ n \quad (n \in \mathbb{Z}) \]
From this, we find:
\[ \theta = 60^\circ + 360^\circ n \]
For \( n = 0 \), we have:
\[ \theta = 60^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 60^\circ \]