9. (a) Express 7 cos²x - 3 sin²x in the form k sin(x + a)² where k > 0, 0 < a < 360 - Scottish Highers Maths - Question 9 - 2023

To express the given equation in the required form, we can use the compound angle formula for sine:

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

We will first rewrite the expression in terms of sin²x and cos²x: $ext{cos}^{2}x = 1 - ext{sin}^{2}x$. Thus, $7 ext{cos}^{2}x - 3 ext{sin}^{2}x = 7(1 - ext{sin}^{2}x) - 3 ext{sin}^{2}x = 7 - 10 ext{sin}^{2}x$

Next, we want to rewrite this in the form $k ext{sin}(x + a)^{2}$. We compare coefficients and find:

1. Set: $k ext{cos}^{2}a = -10$
2. Set: $k ext{sin}^{2}a = 7$

From these equations, we can derive: k = rac{7}{ ext{sin}^{2}a}, leading to ext{sin}^{2}a = rac{7}{k}. Using the Pythagorean identity, we can solve for $k$ and $a$.