Express 2cosx - 5sinx in the form Rcos(x + θ), where R > 0 and 0° < θ < 90°. Hence, or otherwise, solve the equation 2cos2x − 5sin2x = 2.5 for 0° ≤ x ≤ 360°. Give values of x between 0° and 360° at which the maximum and minimum values of 2cos2x - 5sin2x occur.

Comparing this to $2\cos x - 5\sin x$, we equate the coefficients of $\cos x$ and $\sin x$:

Square both equations (1) and (2) and add them.

Divide equation (2) by equation (1).

Rounding to one decimal place, $\theta \approx 68.2^\circ$. This value satisfies $0^\circ < \theta < 90^\circ$.
Therefore, $2\cos x - 5\sin x = \boxed{\sqrt{29}\cos(x + 68.2^\circ)}$.

Substitute this into the given equation, using the more precise value of $\theta \approx 68.19859^\circ$:

Given $0^\circ \le x \le 360^\circ$, then $0^\circ \le 2x \le 720^\circ$.
Adding $\theta \approx 68.19859^\circ$ to all parts:

Let $Y = 2x + \theta$.

The principal value is $Y_1 = \arccos\left(\frac{2.5}{\sqrt{29}}\right) \approx 62.3538^\circ$.

Since $\cos Y$ is positive, $Y$ is in the first or fourth quadrant. The general solutions are $Y = Y_1 + n \cdot 360^\circ$ and $Y = 360^\circ - Y_1 + n \cdot 360^\circ$.

• For $Y = 62.3538^\circ + n \cdot 360^\circ$: For $n=1$: $Y = 62.3538^\circ + 360^\circ = 422.3538^\circ$. For $n=2$: $Y = 62.3538^\circ + 720^\circ = 782.3538^\circ$.
• For $Y = 360^\circ - 62.3538^\circ + n \cdot 360^\circ = 297.6462^\circ + n \cdot 360^\circ$: For $n=0$: $Y = 297.6462^\circ$. For $n=1$: $Y = 297.6462^\circ + 360^\circ = 657.6462^\circ$. The values for $2x + \theta$ are $297.6462^\circ, 422.3538^\circ, 657.6462^\circ, 782.3538^\circ$.

• $2x + 68.19859^\circ = 297.6462^\circ \implies 2x = 229.44761^\circ \implies x = 114.7238^\circ \approx 114.7^\circ$.
• $2x + 68.19859^\circ = 422.3538^\circ \implies 2x = 354.15521^\circ \implies x = 177.0776^\circ \approx 177.1^\circ$.
• $2x + 68.19859^\circ = 657.6462^\circ \implies 2x = 589.44761^\circ \implies x = 294.7238^\circ \approx 294.7^\circ$.
• $2x + 68.19859^\circ = 782.3538^\circ \implies 2x = 714.15521^\circ \implies x = 357.0776^\circ \approx 357.1^\circ$. The solutions for $x$ are $\boxed{114.7^\circ, 177.1^\circ, 294.7^\circ, 357.1^\circ}$.

The expression is $2\cos 2x - 5\sin 2x = \sqrt{29}\cos(2x + \theta)$.
The maximum value of $\cos(Z)$ is 1, and the minimum value is $-1$.
Thus, the maximum value of the expression occurs when $\cos(2x + \theta) = 1$.
The minimum value of the expression occurs when $\cos(2x + \theta) = -1$.

Set $\cos(2x + \theta) = 1$. The general solution for $\cos Z = 1$ is $Z = n \cdot 360^\circ$.
We need $2x + \theta = n \cdot 360^\circ$ within the range $68.19859^\circ \le 2x + \theta \le 788.19859^\circ$.

• For $n=1$: $2x + 68.19859^\circ = 360^\circ \implies 2x = 291.80141^\circ \implies x = 145.9007^\circ \approx 145.9^\circ$.
• For $n=2$: $2x + 68.19859^\circ = 720^\circ \implies 2x = 651.80141^\circ \implies x = 325.9007^\circ \approx 325.9^\circ$. The values of $x$ for the maximum value are $\boxed{145.9^\circ, 325.9^\circ}$.

Set $\cos(2x + \theta) = -1$. The general solution for $\cos Z = -1$ is $Z = 180^\circ + n \cdot 360^\circ$.
We need $2x + \theta = 180^\circ + n \cdot 360^\circ$ within the range $68.19859^\circ \le 2x + \theta \le 788.19859^\circ$.

• For $n=0$: $2x + 68.19859^\circ = 180^\circ \implies 2x = 111.80141^\circ \implies x = 55.9007^\circ \approx 55.9^\circ$.
• For $n=1$: $2x + 68.19859^\circ = 180^\circ + 360^\circ = 540^\circ \implies 2x = 471.80141^\circ \implies x = 235.9007^\circ \approx 235.9^\circ$. The values of $x$ for the minimum value are $\boxed{55.9^\circ, 235.9^\circ}$.