6. f(x) = 12 cos x - 4 sin x - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 5

We rewrite the equation:

$R \cos(x + α) = 7$
where $R \approx 12.6$ and $α \approx 18.43°$.

Now we have:

$\cos(x + α) = \frac{7}{12.6} \approx 0.5534$

This gives:

$x + α = \cos^{-1}(0.5534) \\ x + 18.43° = 56.4° \\ \Rightarrow x = 56.4° - 18.43° \approx 37.97°$

And for the second solution:

$x + α = 360° - 56.4° \\ x + 18.43° = 303.6° \\ \Rightarrow x = 303.6° - 18.43° \approx 285.17°$

Thus the solutions for 0 ≤ x ≤ 360° are approximately:

$x \approx 38.0°, 285.2°$

The minimum value of the expression (12 \cos x - 4 \sin x) occurs at (-R), which is:

$\text{Minimum value} = -\sqrt{160} \approx -12.6$

The minimum occurs when:

$\cos(x + α) = -1$

This gives:

$x + α = 180° \\ x = 180° - 18.43° \approx 161.57°$

Thus, the smallest positive value of x is approximately:

$x = 161.57°$