From Förberedande kurs i matematik 1

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

\displaystyle \sin \left( \frac{\pi }{3}+v \right)=\sin \frac{\pi }{3}\centerdot \cos v+\cos \frac{\pi }{3}\centerdot \sin v.

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

\displaystyle \sin \left( \frac{\pi }{3}+v \right)=\frac{\sqrt{3}}{2}\sqrt{1-a^{2}}+\frac{1}{2}a.