Aus Online Mathematik Brückenkurs 1

If we consider the entire expression \displaystyle x + 40^{\circ} as an unknown, we have a basic 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+40^{\circ}\le 360^{\circ} namely \displaystyle x+40^{\circ} = 65^{\circ} and the symmetric solution \displaystyle x + 40^{\circ} = 180^{\circ} - 65^{\circ} = 115^{\circ}\,.

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

\displaystyle x + 40^{\circ} = 65^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x + 40^{\circ} = 115^{\circ} + n\cdot 360^{\circ}

for all integers n, which gives

\displaystyle x = 25^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x=75^{\circ} + n\cdot 360^{\circ}\,\textrm{.}