Answer: The trick is to nest an aligned environment within the align* one.
Background
Indeed, you want a local alignment (the two-line equality), within a global alignment (flushed right). The aligned environment created a fixed block, that is itself positioned relatively to the other lines.
(As explained here align* environment creates a math-mode. So when you already are in a math-mode, you should then use an aligned-environment.)
In both align* and aligned environments, you must write your equations on separated line. You indicate that a new line starts with \\. Then, to know how/where to align these lines, these environments use the 'key' &. They flush-right or flush-left the as following:
right flushed text & left flushed text & right flushed text & left flushed text \\
loooooooong text that defines the alignment & some text & some text & lorem ipsum \\
lorem & some other text & always the same story about lengths & lorem ipsum \\
some text & loooooooong text that defines the alignment & lorem & some other text \\
MWE

\documentclass{scrartcl}
\usepackage{amsmath}
\begin{document}
\begin{align*}
2\sin2\theta+1=0 &\\
\sin2\theta=-\dfrac{1}{2} &\\
\begin{aligned}
\alpha &=\sin^{-1}\left(\dfrac{1}{2}\right)\\
&=\dfrac{\pi}{6}
\end{aligned} &\\
2\theta=\pi+\alpha,2\pi-\alpha,3\pi+\alpha,4\pi-\alpha &\\
\end{align*}
\end{document}