8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ − a), where R and a are constants, R > 0 and 0 < a < 90° - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 5

The maximum value of the expression occurs when: $\cos(θ - a) = 1$

This gives us: $θ - a = 0 \\Rightarrow θ = a = 53°$

Thus, the maximum value of 3 cos θ + 4 sin θ is: $5$

And the smallest positive value of θ for which this maximum occurs is: $53°$

We need to find the minimum temperature using the given function:

$f(t) = 10 + 3 \cos(15t) + 4 \sin(15t)$

To find the minimum, we consider the terms involving cosine and sine. The minimum occurs when: $\cos(15t - α) = -1$ where α is determined by: $tan(α) = \frac{4}{3}$ Thus, the minimum temperature is: $f(t) = 10 + 3(-1) = 10 - 3 = 7°$

From the previous calculation, we know: $15t - α = 180°$ Therefore: $15t = 180° + α$ Solving for t: $t = \frac{180° + 53°}{15} = \frac{233°}{15} = 15.53$

Thus, the minimum temperature occurs at: $t ≈ 15.53$ hours (from midday).