3. (a) Express $4 \, ext{sin} \, x + 5 \, ext{cos} \, x$ in the form $k \, ext{sin}(x + \, ext{a})$ where $k > 0$ and $0 < \alpha < 2\pi$ - Scottish Highers Maths - Question 3 - 2022

To express the function $4 \text{sin} x + 5 \text{cos} x$ in the form $k \text{sin}(x + \alpha)$, we will use the compound angle formula:

$k \text{sin}(x + \alpha) = k (\text{sin} x \cos \alpha + \text{cos} x \sin \alpha)$

Setting the coefficients equal gives us the equations:

• $k \cos \alpha = 4$
• $k \sin \alpha = 5$

We can find $k$ by using the Pythagorean identity:

$k = \sqrt{(k \cos \alpha)^2 + (k \sin \alpha)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}.$

Now, we can express $k$ as: $k = \sqrt{41}$

Next, we find the angles. Using the previously established equations, we have:

$\cos \alpha = \frac{4}{\sqrt{41}} \quad \text{and} \quad \sin \alpha = \frac{5}{\sqrt{41}}$

Thus, we can express $4 \text{sin} x + 5 \text{cos} x$ as: $$ \sqrt{41} \text{sin}\left(x + \alpha \right) \text{ where } \alpha = \tan^{-1}\left(\frac{5}{4}\right) $.