Express $3 \, ext{sin} \, x + 2 \, ext{cos} \, x$ in the form $R \, ext{sin}(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 7 - 2007 - Paper 5

The greatest value of $3 \, \text{sin} \, x + 2 \, \text{cos} \, x$ occurs when $\sin(x + 0.588) = 1$.
This gives: $3 \, \text{sin} \, x + 2 \, \text{cos} \, x = \sqrt{13}.$
The greatest value of $(3 \, \text{sin} \, x + 2 \, \text{cos} \, x)^4$ is then: $\left(\sqrt{13}\right)^4 = 169.$

To solve the equation $3 \, \text{sin} \, x + 2 \, \text{cos} \, x = 1$, we first rewrite it using the expression found in part (a):
$\sqrt{13} \, \text{sin}(x + 0.588) = 1.$
Then, we can determine: $\text{sin}(x + 0.588) = \frac{1}{\sqrt{13}}.$
Using a calculator, we find: $x + 0.588 = \arcsin(\frac{1}{\sqrt{13}}).$
Calculating gives: $x + 0.588 \approx 0.281 \quad \text{or} \quad \pi - 0.281 \approx 2.860.$
Thus:

• For $x + 0.588 = 0.281$:
  $x \approx 0.281 - 0.588 \approx -0.307$ (not valid in the range)
• For $x + 0.588 = 2.860$:
  $x \approx 2.860 - 0.588 \approx 2.272.$
  Also considering periodic solutions: $x + 0.588 \approx 2.860 + 2\pi \Rightarrow x \approx 2.272 + 2\pi \approx 5.855.$
  Consolidating our answers, we have: $x \approx 2.272, 5.855$ (to 3 decimal places).