Express $2\cos\theta + 9\sin\theta$ in the form of $R\cos(\theta - \alpha)$ where $R > 0$ and $0 < \alpha < \dfrac{\pi}{2}$, giving the values of $R$ and $\alpha$. Hence

1. Calculate the values of $R$ and $\alpha$ required for the substitution (from the prior un-asked part)

To express $2\cos\theta + 9\sin\theta$ in the form $R\cos(\theta - \alpha)$, we expand to get $R\cos\theta\cos\alpha + R\sin\theta\sin\alpha$.
By comparing coefficients:
$R\cos\alpha = 2$
$R\sin\alpha = 9$
$R = \sqrt{2^2 + 9^2} = \sqrt{4 + 81} = \sqrt{85}$
$\tan\alpha = \frac{9}{2} \implies \alpha = \tan^{-1}(4.5) \approx 1.3521 \text{ rad}$

2. Determine the maximum value of the expression

The expression can now be written as $\sqrt{85}\cos(\theta - 1.3521)$.
The maximum value of the cosine function, $\cos(\theta - \alpha)$, is $1$.
Therefore, the maximum value of the expression is $\sqrt{85} \times 1 = \sqrt{85}$.

3. Find the smallest positive value of $\theta$ at which the maximum occurs

The maximum occurs when the cosine term equals $1$, which means:
$\cos(\theta - 1.3521) = 1$
The smallest non-negative angle yielding a cosine of $1$ is $0$.
$\theta - 1.3521 = 0$
$\theta = 1.3521$
$\theta \approx 1.35 \text{ rad (to 3 s.f.)}$

1. Calculate the parameters $R$ and $\alpha$ required for the harmonic form substitution.

To express $2\cos\theta + 9\sin\theta$ in the form $R\cos(\theta - \alpha)$, we find $R$ and $\alpha$:
$R = \sqrt{2^2 + 9^2} = \sqrt{4 + 81} = \sqrt{85}$
$\cos\alpha = \frac{2}{\sqrt{85}}, \sin\alpha = \frac{9}{\sqrt{85}} \implies \tan\alpha = \frac{9}{2} = 4.5$
Since $0 < \alpha < \frac{\pi}{2}$,
$\alpha = \tan^{-1}(4.5) \approx 1.3521 \text{ rad}$

2. Rewrite the equation using the harmonic form and find the reference angle.

Substitute the harmonic form into the target equation:
$\sqrt{85}\cos(\theta - 1.3521) = 1$
$\cos(\theta - 1.3521) = \frac{1}{\sqrt{85}}$
The reference (basic) angle is:
$\cos^{-1}\left(\frac{1}{\sqrt{85}}\right) \approx 1.4621 \text{ rad}$

3. Solve for the smallest positive value of $\theta$.

Considering the possible quadrants where cosine is positive:
$\theta - 1.3521 = \pm 1.4621$
Case 1 (Positive branch):
$\theta = 1.3521 + 1.4621 = 2.8142$
Case 2 (Negative branch):
$\theta = 1.3521 - 1.4621 = -0.1100 \text{ (reject as negative)}$
The next positive root from Case 2 would be $-0.1100 + 2\pi \approx 6.173$, so the smallest positive solution is $2.8142$.
Rounding to 3 significant figures:
$\theta = 2.81 \text{ rad}$