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 3 - 2013 - Paper 8

We start by setting the equation:

$7cos x + sin x = 5$

Dividing both sides by R:

cos(x - α) = rac{5}{R} = rac{5}{ ext{sqrt}(50)}

Calculating gives:

cos(x - 8.1°) = rac{5}{5 ext{surd}2} = rac{5 ext{surd}2}{10} = 0.35355

Now, finding the angle:

1. $x - 8.1° = cos^{-1}(0.35355)$ gives $x ≈ 53.1° + 8.1° = 61.2°$
2. Using the cosine property, $x - 8.1° = 360° - 53.1°$ gives $x ≈ 323.1°$

Thus, the solutions are: $x ≈ 61.2°, 323.1°$.

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

1. Finding the maximum of $f(x) = 7cos x + sin x$:

   • The maximum occurs occurs when tan x = rac{1}{7}
   • The value is $k_{max} = 7 ext{surd}2$.

2. Finding the minimum of $f(x) = 7cos x + sin x$:

   • The minimum occurs when $tan x = -7$
   • The value is $k_{min} = - ext{surd}50$.

Thus, the values of k for which there is only one solution are:

$k = 7 ext{surd}2, - ext{surd}50$.