Let non-zero vectors such that $(\mathbf{a}\times \mathbf{b})\times \mathbf{c}=\frac{1}{3}|\mathbf{b}\parallel \mathbf{c}|\mathbf{a}$ If $\theta$ is the acute angle between the vectors$b and c$, then sin$\theta$ equals: 

$(\mathbf{a}\times \mathbf{b})\times \mathbf{c}=\frac{1}{3}|\mathbf{b}\parallel \mathbf{c}|\mathbf{a}$

$\begin{array}{l}\Rightarrow (\mathbf{a}\cdot \mathbf{c})\mathbf{b}-(\mathbf{b}\cdot \mathbf{c})\mathbf{a}=\frac{1}{3}|\mathbf{b}\parallel \mathbf{c}|\mathbf{a}\\ \Rightarrow (\mathbf{a}\cdot \mathbf{c})\mathbf{b}=\left\{(\mathbf{b}\cdot \mathbf{c})+\frac{1}{3}|\mathbf{b}\parallel \mathbf{c}|\right\}\mathbf{a}\\ \Rightarrow (\mathbf{a}\cdot \mathbf{c})\mathbf{b}=|\mathbf{b}\parallel \mathbf{c}|\left\{\cos \theta +\frac{1}{3}\right\}\mathbf{a}\end{array}$

As a and b are not parallel, $\mathbf{a}\cdot \mathbf{c}=0\text{ and }\cos \theta +\frac{1}{3}=0$

$\begin{array}{l}\Rightarrow \cos \theta =-\frac{1}{3}\\ \Rightarrow \sin \theta =\frac{2\sqrt{2}}{3}\end{array}$