Aus Online Mathematik Brückenkurs 1

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

\displaystyle \cos^2 v + \sin^2 v = 1\qquad\Leftrightarrow\qquad \cos v = \pm\sqrt{1-\sin^2 v}\,\textrm{.}

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 x-coordinate (cosine value); thus, we can conclude that

\displaystyle \cos v = \sqrt{1-\sin^2 v} = \sqrt{1-a^2}\,\textrm{.}