The solution set of the system of equations `x+y=(2pi)/3,cosx+cosy=3/2,` where `xa n dy` are real, is ________

Explanation

To solve the system of equations given by:

1. \( x + y = \frac{2\pi}{3} \)
2. \( \cos x + \cos y = \frac{3}{2} \)

we will follow these steps:

### Step 1: Use the first equation to express \( y \) in terms of \( x \).
From the first equation, we can express \( y \) as:
\[ y = \frac{2\pi}{3} - x \]

### Step 2: Substitute \( y \) into the second equation.
Now, we substitute \( y \) into the second equation:
\[ \cos x + \cos\left(\frac{2\pi}{3} - x\right) = \frac{3}{2} \]

### Step 3: Use the cosine subtraction formula.
Using the cosine subtraction formula, we know that:
\[ \cos\left(\frac{2\pi}{3} - x\right) = \cos\frac{2\pi}{3} \cos x + \sin\frac{2\pi}{3} \sin x \]
where \( \cos\frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \). Thus, we can rewrite the equation as:
\[ \cos x - \frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x = \frac{3}{2} \]

### Step 4: Simplify the equation.
This simplifies to:
\[ \frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x = \frac{3}{2} \]

### Step 5: Multiply through by 2 to eliminate the fraction.
Multiplying the entire equation by 2 gives:
\[ \cos x + \sqrt{3} \sin x = 3 \]

### Step 6: Analyze the range of the left-hand side.
The expression \( \cos x + \sqrt{3} \sin x \) can be rewritten in the form \( R \cos(x - \phi) \), where \( R = \sqrt{1^2 + (\sqrt{3})^2} = 2 \) and \( \phi = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \). Thus, we have:
\[ \cos x + \sqrt{3} \sin x = 2 \cos\left(x - \frac{\pi}{3}\right) \]
The maximum value of \( 2 \cos\left(x - \frac{\pi}{3}\right) \) is 2, which occurs when \( \cos\left(x - \frac{\pi}{3}\right) = 1 \).

### Step 7: Conclusion about the solution set.
Since the left-hand side can only take values in the range \([-2, 2]\) and we have \( \cos x + \sqrt{3} \sin x = 3\), which is outside this range, we conclude that:
\[ \text{The system of equations has no solution.} \]

### Final Answer:
The solution set of the system of equations is **empty**.

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