Let 12 cos θ + 5 sin θ = R cos (θ - α), where R > 0 and α is an acute angle to be found.

Here, a = 12 and b = 5.

R = sqrt (12² + 5²)
R = 13

α = inverse tangent (5/12)
α = 22.61986495°

12 cos θ + 5 sin θ = 13 cos (θ - 22.6°)

For our expression in part (d), the value of the fraction is maximised when the denominator is minimised to a positive value which is close to zero.

Denominator = 15 + 13 cos (θ - 22.6°)

Since -1 ≤ cos (θ - 22.6°) ≤ 1, the smallest possible positive value of the denominator
= 15 + 13 (-1)
= 2

So, the greatest possible value of the fraction is 1/2.

This occurs when cos (θ - 22.6°) = -1.

θ - 22.6° = 180° (one value only)
θ = 202.6°