f(x) = 7cos x + sin x Given that f(x) = Rcos(x - α), where R > 0 and 0 < α < 90°, a) find the exact value of R and the value of α to one decimal place - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 8

We have the equation:

$7\cos x + \sin x = 5$.

Rearranging gives:

$\sqrt{50}\cos\left(x - 8.1°\right) = 5$.

Dividing both sides by ( R = \sqrt{50} ), we have:

$\cos\left(x - 8.1°\right) = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}$.

From this, we can find:

$x - 8.1° = 45° \Rightarrow x = 53.1°$

and

$x - 8.1° = -45° \Rightarrow x = 323.1°$.

For the equation to have only one solution, the value of k must equal the maximum or minimum of the function.

The maximum occurs at:

$R = \sqrt{50} \approx 7.07$

and the minimum occurs at:

$-R = -\sqrt{50} \approx -7.07$.

Thus, the values of k for which the equation has only one solution in the interval 0 ≤ x < 360° are:

$k = \pm \sqrt{50}$.