Aus Online Mathematik Brückenkurs 1

If we consider the entire expression \displaystyle x+\text{4}0^{\circ } as an unknown, we have a fundamental trigonometric equation and can, with the aid of the unit circle, see that there are two solutions to the equation for \displaystyle 0^{\circ }\le x+\text{4}0^{\circ }\le \text{36}0^{\circ } namely \displaystyle x+\text{4}0^{\circ }=\text{65}^{\circ } and the symmetric solution \displaystyle x+\text{4}0^{\circ }=\text{18}0^{\circ }-\text{65}^{\circ }=\text{115}^{\circ }.

It is then easy to set up the general solution by adding multiples of \displaystyle 360^{\circ },

\displaystyle x+\text{4}0^{\circ }=\text{65}^{\circ }+n\centerdot 360^{\circ } and \displaystyle x+\text{4}0^{\circ }=\text{115}^{\circ }+n\centerdot 360^{\circ }

for all integers \displaystyle n, which gives

\displaystyle x=2\text{5}^{\circ }+n\centerdot 360^{\circ } and \displaystyle x=7\text{5}^{\circ }+n\centerdot 360^{\circ }