Express $7 \cos x - 24 \sin x$ in the form $R \cos (x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 4

The minimum value of $7 \cos x - 24 \sin x$ can be derived from the maximum value of the cosine function. Since the range of $\cos$ is between -1 and 1, the minimum value of the expression is: $-R = -25.$

To solve the equation: $7 \cos x - 24 \sin x = 10$
Rewriting it using the previously found expression: $25 \cos (x + 1.287) = 10$
Simplifying gives us: $\cos (x + 1.287) = \frac{10}{25} = 0.4.$
Now, find $x + 1.287$: $x + 1.287 = \cos^{-1}(0.4) \approx 1.159, \text{ or } -1.159.$
Thus, $x$ values:

1. For $x + 1.287 = 1.159$, we have: $x \approx 1.159 - 1.287 \approx -0.128 \Rightarrow \text{(Not valid for this range)}$
2. For the second value, apply: $x + 1.287 = 2\pi - 1.159 \implies x \approx 6.124 - 1.287 \approx 4.837.$
3. Another solution: $x + 1.287 = 2\pi + 1.159 \implies x \approx 2\pi + 1.159 - 1.287$ Hence obtaining:

• Valid solutions are approximately: $x \approx 4.84 \text{ and } x \approx 1.36$ Therefore, the solutions to two decimal places are: $x \approx 4.84 \text{ and } x \approx 1.36.$