From Förberedande kurs i matematik 1

We start by subtracting \displaystyle 2\pi from \displaystyle \frac{11\pi }{3}, so that we get an angle between \displaystyle o and \displaystyle 2\pi . This doesn't change the cosine value

\displaystyle \cos \frac{11\pi }{3}=\cos \left( \frac{11\pi }{3}-2\pi \right)=\cos \frac{5\pi }{3}

Then, by rewriting \displaystyle \frac{5\pi }{3} as a sum of \displaystyle \pi - and \displaystyle \frac{\pi }{2} -terms

\displaystyle \frac{5\pi }{3}=\frac{3\pi +\frac{3}{2}\pi +\frac{1}{2}\pi }{3}=\pi +\frac{\pi }{2}+\frac{\pi }{6}

we see that \displaystyle \frac{5\pi }{3} is an angle in the fourth quadrant which makes an angle \displaystyle \frac{\pi }{6} with the negative \displaystyle y -axis.

With the help of a triangle and a little trigonometry, we can determine the coordinates for the point on a unit circle which corresponds to the angle \displaystyle \frac{5\pi }{3} .

The point has coordinates \displaystyle \left( \frac{1}{2} \right.,\left. -\frac{\sqrt{3}}{2} \right) and \displaystyle \cos \frac{11\pi }{3}=\cos \frac{5\pi }{3}=\frac{1}{2}.