Answer
$$\cos \left( {\frac{{7\pi }}{3}} \right) = \frac{1}{{\text{2}}}$$
Work Step by Step
$$\eqalign{
& \cos \left( {\frac{{7\pi }}{3}} \right) \cr
& \frac{{7\pi }}{3} = 2\pi + \frac{\pi }{3} \cr
& {\text{then}} \cr
& \cos \left( {\frac{{7\pi }}{3}} \right) = \cos \left( {\frac{\pi }{3}} \right) \cr
& {\text{convert }}\frac{\pi }{3}{\text{radians to degrees}} \cr
& \frac{\pi }{3}{\text{radians}} = \frac{\pi }{3}\left( {\frac{{{{180}^ \circ }}}{\pi }} \right) \cr
& \frac{\pi }{3}{\text{radians}} = {60^ \circ } \cr
& \cos {60^ \circ } = \frac{{{\text{adjacent side to the 6}}{{\text{0}}^ \circ }}}{{{\text{hyppotenuse}}}} \cr
& {\text{using the }}{30^ \circ }{\text{ - 6}}{{\text{0}}^ \circ }{\text{ - 9}}{{\text{0}}^ \circ }{\text{ triangle to obtain}} \cr
& \cos {60^ \circ } = \frac{1}{{\text{2}}} \cr
& {\text{then }} \cr
& \cos \left( {\frac{{7\pi }}{3}} \right) = \frac{1}{{\text{2}}} \cr} $$
