Aus Online Mathematik Brückenkurs 1

An often-used technique to calculate \displaystyle \cos v and \displaystyle \tan v, given the sine value of an acute angle, is to draw the angle \displaystyle v in a right-angled triangle which has two sides arranged so that \displaystyle \sin v = 5/7\,.

Using the Pythagorean theorem, we can determine the length of the third side in the triangle.

\displaystyle \begin{align}&x^2 + 5^2 = 7^2\\[5pt] &\text{which gives that}\\[5pt] &x = \sqrt{7^2-5^2} = \sqrt{24} = 2\sqrt{6}\end{align}

Then, using the definition of cosine and tangent,

\displaystyle \begin{align} \cos v &= \frac{x}{7} = \frac{2\sqrt{6}}{7}\,,\\[5pt] \tan v &= \frac{5}{x} = \frac{5}{2\sqrt{6}}\,\textrm{.} \end{align}

Note: The right-angled triangle that we use is just a tool and has nothing to do with the triangle that is referred to in the question.