General solution of the equation
`(sqrt(3) - 1) sin theta + (sqrt(3) + 1) cos theta = 2` is

Options

A. `2 n pi +- (pi//4) + (pi//12)`
B. `npi + (-1)^(n) cdot (pi//4)+(pi//12)`
C. `2npi +- (pi//4) - (pi//12)`
D. `npi + (-1)^(n) cdot (pi//4) - (pi//12)`

Answer

To find the general solution of the equation

\[ (\sqrt{3} - 1) \sin \theta + (\sqrt{3} + 1) \cos \theta = 2, \]

we will follow these steps:

### Step 1: Normalize the equation
We start by dividing the entire equation by \(2\sqrt{2}\):

\[ \frac{(\sqrt{3} - 1)}{2\sqrt{2}} \sin \theta + \frac{(\sqrt{3} + 1)}{2\sqrt{2}} \cos \theta = \frac{2}{2\sqrt{2}}. \]

This simplifies to:

\[ \frac{\sqrt{3} - 1}{2\sqrt{2}} \sin \theta + \frac{\sqrt{3} + 1}{2\sqrt{2}} \cos \theta = \frac{1}{\sqrt{2}}. \]

### Step 2: Identify sine and cosine values
Next, we recognize that:

\[ \frac{\sqrt{3} - 1}{2\sqrt{2}} = \sin 15^\circ \quad \text{and} \quad \frac{\sqrt{3} + 1}{2\sqrt{2}} = \cos 15^\circ. \]

Thus, we can rewrite the equation as:

\[ \sin 15^\circ \sin \theta + \cos 15^\circ \cos \theta = \frac{1}{\sqrt{2}}. \]

### Step 3: Use the cosine addition formula
Using the cosine addition formula, we can express the left side as:

\[ \cos(\theta - 15^\circ) = \frac{1}{\sqrt{2}}. \]

### Step 4: Solve for \(\theta\)
Now, we solve for \(\theta\):

\[ \theta - 15^\circ = 45^\circ + 360^\circ n \quad \text{or} \quad \theta - 15^\circ = -45^\circ + 360^\circ n, \]

where \(n\) is any integer.

### Step 5: Rearranging the equations
From the first equation:

\[ \theta = 60^\circ + 360^\circ n. \]

From the second equation:

\[ \theta = -30^\circ + 360^\circ n. \]

### Step 6: General solution
Thus, the general solution can be expressed as:

\[ \theta = 60^\circ + 360^\circ n \quad \text{or} \quad \theta = -30^\circ + 360^\circ n. \]

### Final Answer:
The general solution of the equation is:

\[ \theta = 60^\circ + 360^\circ n \quad \text{or} \quad \theta = -30^\circ + 360^\circ n, \quad n \in \mathbb{Z}. \]

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