Expression Maximization

We need to find the value of \(x\) that maximizes the expression \(E = \cos x + \sqrt{3} \sin x\). This expression is of the form \(a \cos x + b \sin x\), where \(a = 1\) and \(b = \sqrt{3}\).

Trigonometric Transformation

We can rewrite the expression in the form \(R \cos(x - \alpha)\) or \(R \sin(x + \alpha)\). Let's use the form \(R \cos(x - \alpha)\).

1. Calculate \(R\): \(R = \sqrt{a^2 + b^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2\).
2. Rewrite the expression: \(E = 2 \left( \frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x \right)\).
3. Identify the angle \(\alpha\): We know that \(\cos(\pi/3) = 1/2\) and \(\sin(\pi/3) = \sqrt{3}/2\). Substitute these values: \(E = 2 (\cos(\pi/3) \cos x + \sin(\pi/3) \sin x)\).
4. Apply the cosine difference identity \(\cos(A - B) = \cos A \cos B + \sin A \sin B\): \(E = 2 \cos(x - \pi/3)\).

Finding the Maximum

The maximum value of the cosine function, \(\cos(\theta)\), is 1.

Therefore, the expression \(E = 2 \cos(x - \pi/3)\) is maximum when \(\cos(x - \pi/3) = 1\).

Solving for x

The equation \(\cos(x - \pi/3) = 1\) holds true when the argument \((x - \pi/3)\) is an integer multiple of \(2\pi\). That is, \(x - \pi/3 = 2n\pi\), where \(n\) is an integer.

For the principal value, we consider \(n=0\): \(x - \pi/3 = 0\).

Solving for \(x\): \(x = \pi/3\).

Thus, the expression \(\cos x + \sqrt{3} \sin x\) is maximum when \(x = \pi/3\).