Roots, $\sin \theta$ and $\cos \theta$

Relevant wiki: Vieta's Formula Problem Solving - Intermediate

If $\sin \theta$ and $\cos \theta$ are roots of $3x^2 - x + k = 0$ , then by Vieta's formula:

$\begin{cases} \sin \theta + \cos \theta = \dfrac 13 & ...(1) \\ \sin \theta \cos \theta = \dfrac k3 & ...(2) \end{cases}$

And we have:

$\begin{aligned} (\sin \theta + \cos \theta)^2 & = \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta \\ \frac 19 & = 1 + \frac {2k}3 \\ \implies k & = - \frac 43 \end{aligned}$

Also:

$\begin{aligned} \sin^3 \theta + \cos^3 \theta & = (\sin \theta + \cos \theta)(\sin^2 \theta + \cos^2 \theta - \sin \theta \cos \theta) \\ & = \frac 13 \left(1 + \frac 49\right) = \frac {13}{27} \end{aligned}$

$\implies 54(\sin^3 \theta + \cos^3 \theta) = \boxed {26}$ .