If
\[
\alpha=\tan\left(2\sin^{-1}\left(\frac23\right)\right)
\quad\text{and}\quad
\beta=\sin\left(2\tan^{-1}\left(\frac13\right)\right),
\]
then the maximum value of
\[
\alpha\sin\theta+\beta\cos\theta
\]
is
Show Hint
The maximum value of
\[
a\sin\theta+b\cos\theta
\]
is always
\[
\boxed{\sqrt{a^2+b^2}}.
\]
First evaluate the constants \(a\) and \(b\), then apply this standard result.