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

We need to solve:

$12 cos x - 4 sin x = 7$

Rearranging gives us:

$12 cos x - 4 sin x - 7 = 0$

In standard form:

Letting R = 4√10 and using the expression:

$R cos(x + α) = 7$

This yields:

$cos(x + α) = \frac{7}{R} = \frac{7}{4\\sqrt{10}} \approx 0.5534$

We find x + α:

$x + α = \arccos(0.5534) = 56.4° \, \text{(principle value)}$

Thus, we solve for x:

• First solution: $x ≈ 56.4° - α ≈ 56.4° - 18.43° = 38.0°$

• The second solution can be found using: $x + α = 360° - 56.4° = 303.6°$ Taking this into account: $x = 303.6° - 18.43° = 285.2°$

Finally, the answers are:

• First solution: 38.0°
• Second solution: 285.2°

The minimum value of 12 cos x - 4 sin x can be established by evaluating:

$-R = -\sqrt{160} = -4\\sqrt{10} \approx -12.6$

The minimum value occurs when:

$cos(x + α) = -1$

This corresponds to:

$x + α = 180° \, \text{(since }\cos(180°) = -1\text{)}$

Thus, solving gives: $x = 180° - α ≈ 180° - 18.43° ≈ 161.57°$

Rounding to two decimal places, we find:

$x ≈ 161.57°$