If `4cos^2 theta + sqrt 3 = 2[sqrt3 + 1] cos theta,` then `theta` =

Options

A. `2n pi pm pi/3`
B. `2n pi pm pi/6`
C. `2n pi pm pi/4`
D. `2npi`

Answer

To solve the equation \( 4\cos^2 \theta + \sqrt{3} = 2(\sqrt{3} + 1) \cos \theta \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ 4\cos^2 \theta + \sqrt{3} = 2(\sqrt{3} + 1) \cos \theta \]
Expanding the right-hand side:
\[ 2(\sqrt{3} + 1) \cos \theta = 2\sqrt{3} \cos \theta + 2 \cos \theta \]
Now, we can rewrite the equation as:
\[ 4\cos^2 \theta + \sqrt{3} - 2\sqrt{3} \cos \theta - 2 \cos \theta = 0 \]
This simplifies to:
\[ 4\cos^2 \theta - (2\sqrt{3} + 2) \cos \theta + \sqrt{3} = 0 \]

### Step 2: Setting Up the Quadratic Equation
Now we have a quadratic equation in terms of \( \cos \theta \):
\[ 4\cos^2 \theta - (2\sqrt{3} + 2) \cos \theta + \sqrt{3} = 0 \]
Let \( x = \cos \theta \). The equation becomes:
\[ 4x^2 - (2\sqrt{3} + 2)x + \sqrt{3} = 0 \]

### Step 3: Using the Quadratic Formula
To solve for \( x \), we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 4 \), \( b = -(2\sqrt{3} + 2) \), and \( c = \sqrt{3} \).

Calculating \( b^2 - 4ac \):
\[ b^2 = (2\sqrt{3} + 2)^2 = 4(3 + 2\sqrt{3} + 1) = 16 + 8\sqrt{3} \]
\[ 4ac = 4 \cdot 4 \cdot \sqrt{3} = 16\sqrt{3} \]
Thus,
\[ b^2 - 4ac = (16 + 8\sqrt{3}) - 16\sqrt{3} = 16 - 8\sqrt{3} \]

Now substituting back into the quadratic formula:
\[ x = \frac{2\sqrt{3} + 2 \pm \sqrt{16 - 8\sqrt{3}}}{8} \]

### Step 4: Simplifying the Roots
We will find the roots:
1. \( x_1 = \frac{2\sqrt{3} + 2 + \sqrt{16 - 8\sqrt{3}}}{8} \)
2. \( x_2 = \frac{2\sqrt{3} + 2 - \sqrt{16 - 8\sqrt{3}}}{8} \)

### Step 5: Finding \( \theta \)
Now, we need to find \( \theta \) from \( \cos \theta = x_1 \) and \( \cos \theta = x_2 \).

Using the identity:
\[ \theta = \cos^{-1}(x) \quad \text{and} \quad \theta = 2\pi n \pm \cos^{-1}(x) \quad (n \in \mathbb{Z}) \]

### Step 6: Final Values of \( \theta \)
After calculating \( x_1 \) and \( x_2 \), we can find the corresponding angles \( \theta \).