From Förberedande kurs i matematik 1

With the help of the Pythagorean identity, we can express \displaystyle \cos v in terms of \displaystyle \text{sin }v,

\displaystyle \cos ^{2}v+\sin ^{2}v=1

In addition, we know that the angle \displaystyle v lies between \displaystyle -{\pi }/{2}\; and \displaystyle {\pi }/{2}\;, i.e. either in the first or fourth quadrant, where angles always have a positive \displaystyle x -coordinate (cosine value); thus, we can conclude that

\displaystyle \cos v=\sqrt{1-\text{sin}^{2}\text{ }v}=\sqrt{1-a^{2}}