6. (a) Express $3 \sin x + 2 \cos x$ in the form $R \sin(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 5

The maximum value of $3 \sin x + 2 \cos x$ occurs when:

$R \sin(x + \alpha) = R$

Thus, the maximum value is:

$\sqrt{13}$

Now, to find the greatest value of $(3 \sin x + 2 \cos x)^4$:

$\left( \sqrt{13} \right)^4 = 13^2 = 169$

To solve this equation, we rewrite it as:

$\sqrt{13} \sin\left(x + 0.588\right) = 1$

Next, we simplify to:

$\sin\left(x + 0.588\right) = \frac{1}{\sqrt{13}}\approx 0.277$

Finding the inverse sine gives:

$x + 0.588 = \sin^{-1}(0.277) \text{ or } \pi - \sin^{-1}(0.277)$

Calculating these yields two values:

1. $x + 0.588 \approx 0.281$ leading to $x \approx 0.281 - 0.588 \approx -0.307$ (not in range)

2. $x + 0.588 \approx 2.862$ leading to $x \approx 2.862 - 0.588 \approx 2.274$

3. $x + 0.588 \approx 3.86$ leading to $x \approx 3.86 - 0.588 \approx 3.272$

Thus:

• $x \approx 2.274$ in $[0, 2\pi)$
• $x \approx 3.272$ in $[0, 2\pi)$