9. (a) Express 7 cos² x - 3 sin² x in the form k sin(x + a)² where k > 0, 0 < a < 360 - Scottish Highers Maths - Question 9 - 2023

To express the above equation in the desired form, we can start by using the identity

$ext{sin}(x + a)^2 = ext{sin}^2 x ext{cos}^2 a + ext{cos}^2 x ext{sin}^2 a + 2 ext{sin} x ext{cos} x ext{cos} a ext{sin} a$

Next, we rewrite 7 cos² x - 3 sin² x:

1. Apply the identity:

   • Note: Using the compound angle formula, we have:

   $7 ext{cos}^2 x - 3 ext{sin}^2 x = 7 ext{c} ext{o}^2 x + (-3) ext{s} ext{i}^2 x$

2. We compare coefficients with the terms $k ext{sin}(x + a)^2$. Therefore, we get:

   k = rac{ ext{sqrt}(58)}{7} ext{ and } a = 113.19^ heta

Thus, $k ext{sin}(x + a)^2 = ext{sqrt}(58) ext{sin}(x + 113.19)^{ heta}$.