From Förberedande kurs i matematik 1

In this case, it is perhaps simplest to use the addition formula for sine,

\displaystyle \sin\Bigl(\frac{\pi}{3}+v\Bigr) = \sin\frac{\pi }{3}\cdot \cos v + \cos\frac{\pi}{3}\cdot\sin v\,\textrm{.}

Since \displaystyle \sin (\pi/3) = \sqrt{3}/\!2, \displaystyle \cos (\pi/3) = 1/2, \displaystyle \sin v = a, and \displaystyle \cos v=\sqrt{1-a^2} this can be written as

\displaystyle \sin\Bigl(\frac{\pi}{3}+v\Bigr) = \frac{\sqrt{3}}{2}\sqrt{1-a^2} + \frac{1}{2}a\,\textrm{.}