f(x) = 7 cos 2x - 24 sin 2x Given that f(x) = R cos(2x + α), where R > 0 and 0 < α < 90°; a) find the value of R and the value of α - Edexcel - A-Level Maths Pure - Question 1 - 2012 - Paper 5

We can rewrite the equation using R and α:

$R \cos(2x + α) = 12.5$

Substituting for R and α gives:

$25 \cos(2x + 73.7°) = 12.5$

Dividing both sides by 25:

$\cos(2x + 73.7°) = 0.5$

The general solutions for this are:

$2x + 73.7° = 60° + k(360°) \text{ or } 2x + 73.7° = 300° + k(360°)$

For k = 0,

1. First equation:

ightarrow 2x = -13.7° ext{ (not valid for } 0 ≤ x < 180°)$$ 2. Second equation: $$2x = 300° - 73.7° ightarrow 2x = 226.3° ightarrow x = 113.15°$$ Thus, the only valid solution in the range is: $$x ≈ 113.2°$$

We start with the expression:

$14 \cos^2 x - 48 \sin x \cos x$

This can be rewritten using the double-angle identities:

$= 14 \left( \frac{1 + \cos(2x)}{2} \right) - 24 \sin(2x)$

This simplifies to:

$= 7 + 7 \cos(2x) - 24 \sin(2x)$

Thus, we identify:

a = 7,

b = -24,

c = 7.

The maximum value of the expression occurs when:

$a \cos(2x) + b \sin(2x)$

The maximum value can be found using:

$R = \sqrt{a^2 + b^2}$

Substituting in the values:

$a = 7, b = -24 \Rightarrow R = \sqrt{7^2 + (-24)^2} = \sqrt{625} = 25$

Thus, the maximum value is:

$= 7 + 25 = 32$.