Aus Online Mathematik Brückenkurs 1

Using the unit circle shows that the equation \displaystyle \cos 3x = -1/\!\sqrt{2} has two solutions for \displaystyle 0\le 3x\le 2\pi\,,

\displaystyle 3x = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}\qquad\text{and}\qquad 3x = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\,\textrm{.}

We obtain the other solutions by adding multiples of \displaystyle 2\pi,

\displaystyle 3x = \frac{3\pi}{4} + 2n\pi\qquad\text{and}\qquad 3x = \frac{5\pi}{4} + 2n\pi\,,

i.e.

\displaystyle x = \frac{\pi}{4} + \frac{2}{3}n\pi\qquad\text{and}\qquad x = \frac{5\pi}{12} + \frac{2}{3}n\pi\,,

where n is an arbitrary integer.