To solve the equation \( \cos x - \sqrt{3} \sin x = 2 \), we can approach it by converting the equation into a simpler form using the method of combining sine and cosine.

### Step 1: Rewrite the equation in a simplified form
We rewrite the expression \( \cos x - \sqrt{3} \sin x \) as a single trigonometric function. This can be done by expressing it in the form:

\[
R \cos(x + \alpha)
\]

where \( R \) is the amplitude and \( \alpha \) is a phase shift.

### Step 2: Find \( R \) and \( \alpha \)
We know that:

\[
R \cos(x + \alpha) = A \cos x + B \sin x
\]

where \( A = 1 \) (coefficient of \( \cos x \)) and \( B = -\sqrt{3} \) (coefficient of \( \sin x \)).

To find \( R \), use the formula:

\[
R = \sqrt{A^2 + B^2} = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
\]

Next, to find \( \alpha \), use:

\[
\alpha = \tan^{-1} \left( \frac{B}{A} \right) = \tan^{-1} \left( \frac{-\sqrt{3}}{1} \right)
\]

\[
\alpha = \tan^{-1} (-\sqrt{3}) = -\frac{\pi}{3}
\]

Thus, the equation becomes:

\[
2 \cos(x - \frac{\pi}{3}) = 2
\]

### Step 3: Solve the equation
Now, solve for \( x \):

\[
\cos(x - \frac{\pi}{3}) = 1
\]

The cosine function equals 1 when the angle is \( 0 \), so:

\[
x - \frac{\pi}{3} = 2n\pi \quad \text{(for any integer \( n \))}
\]

Solving for \( x \):

\[
x = \frac{\pi}{3} + 2n\pi
\]

### Final Answer:
The general solution is:

\[
x = \frac{\pi}{3} + 2n\pi, \quad n \in \mathbb{Z}
\]

This means \( x \) is \( \frac{\pi}{3} \) plus any multiple of \( 2\pi \).

Would you like further clarification or details on any step? Here are some related questions to deepen your understanding:

1. How do you combine trigonometric functions using the amplitude-phase form?
2. Can you solve similar equations involving sine and cosine with different coefficients?
3. What is the significance of the general solution in trigonometric equations?
4. How can you verify the solution to the equation using a graph?
5. What are some real-world applications of solving trigonometric equations?

**Tip:** When solving trigonometric equations, always consider both positive and negative solutions for the sine and cosine functions.

Solve the trigonometric equation cos x - √3 sin x = 2 using the amplitude-phase form method. This detailed step-by-step solution covers key concepts in trigonometry, including combining sine and cosine, the cosine addition formula, and solving for the general solution. Ideal for high school students in Grades 10-12.

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