From Förberedande kurs i matematik 1

Using the addition formula for cosine, we can express \displaystyle \cos (v-\pi/3) in terms of \displaystyle \cos v and \displaystyle \sin v,

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

Since \displaystyle \cos v = b and \displaystyle \sin v = \sqrt{1-b^2} we obtain

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