\(\eqalign{
& a)\,\,1 - i\sqrt 3 = 2\left( {{1 \over 2} - {{\sqrt 3 } \over 2}i} \right) = 2\left( {\cos \left( { - {\pi \over 3}} \right) + i\sin \left( { - {\pi \over 3}} \right)} \right);\,\,\,\,\, \cr
& \,\,\,\,\,\,\,\,1 + i = \sqrt 2 \left( {{1 \over {\sqrt 2 }} + {1 \over {\sqrt 2 }}i} \right) = \sqrt 2 \left( {\cos \left( {{\pi \over 4}} \right) + i\sin \left( {{\pi \over 4}} \right)} \right);\, \cr
& \,\,\,\,\,\,\,\,(1 - i\sqrt 3 )(1 + i) = 2\sqrt 2 \left( {{1 \over 2} - {{\sqrt 3 } \over 2}i} \right)\left( {{1 \over {\sqrt 2 }} + {1 \over {\sqrt 2 }}i} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2\sqrt 2 \left( {\cos \left( { - {\pi \over 3}} \right) + i\sin \left( { - {\pi \over 3}} \right)} \right)\left( {\cos {\pi \over 4} + i\sin {\pi \over 4}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2\sqrt 2 \left[ {\cos \left( {{\pi \over 4} - {\pi \over 3}} \right) + i\sin \left( {{\pi \over 4} - {\pi \over 3}} \right)} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2\sqrt 2 \left[ {\cos \left( { - {\pi \over {12}}} \right) + i\sin \left( { - {\pi \over {12}}} \right)} \right];\,\, \cr
& {{1 - i\sqrt 3 } \over {1 + i}} = \sqrt 2 \left[ {\cos \left( { - {\pi \over 3} - {\pi \over 4}} \right) + i\sin \left( { - {\pi \over 3} - {\pi \over 4}} \right)} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;\;\;\, = \sqrt 2 \left[ {\cos \left( { - {7 \over {12}}\pi } \right) + i\sin \left( { - {7 \over {12}}\pi } \right)} \right]; \cr
& b)\,\,2i = 2\left( {\cos {\pi \over 2} + i\sin {\pi \over 2}} \right) \cr
& \,\,\,\,\,\,\,\left( {\sqrt 3 - i} \right) = 2\left( {{{\sqrt 3 } \over 2} - {1 \over 2}i} \right) = 2\left[ {\cos \left( { - {\pi \over 6}} \right) + i\sin \left( { - {\pi \over 6}} \right)} \right]; \cr
& \,\,\,\,\,\,\,2i\left( {\sqrt 3 - i} \right) = 4\left[ {\cos \left( {{\pi \over 2} - {\pi \over 6}} \right) + i\sin \left( {{\pi \over 2} - {\pi \over 6}} \right)} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \;\;\,= 4\left[ {\cos \left( {{\pi \over 3}} \right) + i\sin \left( {{\pi \over 3}} \right)} \right] \cr
& c)\,\,2 + 2i = 2\sqrt 2 \left( {{1 \over {\sqrt 2 }} + {1 \over {\sqrt 2 }}i} \right) = 2\sqrt 2 \left( {\cos {\pi \over 4} + i\sin {\pi \over 4}} \right)\, \cr
& \Rightarrow {1 \over {2 + 2i}} = {1 \over {2\sqrt 2 }}\left[ {\cos \left( { - {\pi \over 4}} \right) + i\sin \left( { - {\pi \over 4}} \right)} \right] \cr
& d)\,z = \,\sin \varphi + i\cos \varphi = \,\cos \left( {{\pi \over 2} - \varphi } \right) + i\sin\left( {{\pi \over 2} - \varphi } \right)(\varphi \in \mathbb R) \cr} \)
