3. (a) Express $5 \, ext{cos} \, x - 3 \, ext{sin} \, x$ in the form $R \, ext{cos}(x + heta)$, where $R > 0$ and $0 < heta < \frac{1}{2} \, ext{π}$ - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 2

Using the result from part (a), we substitute:

$\sqrt{34} \, \text{cos}\left(x + 0.5404\right) = 4$

Dividing both sides by $\sqrt{34}$ gives:

$\text{cos}\left(x + 0.5404\right) = \frac{4}{\sqrt{34}}$

Now, calculate the value:

$\frac{4}{\sqrt{34}} \approx 0.6859$

Finding $x + 0.5404$:

direction 1: $x + 0.5404 = \text{cos}^{-1}(0.6859) \approx 0.8184$

direction 2: $x + 0.5404 = -0.8184 + 2 \pi$

Solving for $x$ yields:

direction 1: $x = 0.8184 - 0.5404 \approx 0.2780$ direction 2: $x = -0.8184 + 2\pi - 0.5404 \approx 5.4654$

Thus, the solutions are:

$x \approx 0.28, 5.47$ (to 2 decimal places).