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Bài 27. Hãy tìm dạng lượng giác của các số phức: \(\overline z \,;\, – z;\,{1 \over {\overline z }};\,kz\,\left( {k \in \mathbb R^*} \right)\) trong mỗi trường hợp sau:
\(a)\,z = r\left( {\cos \varphi + i\sin\varphi } \right)\,\left( {r > 0} \right);\)
\(b)\,z = 1 + \sqrt 3 i.\)
\(\eqalign{ & a)\,\overline z = r\left( {\cos \varphi – i\sin \varphi } \right) = r\left( {\cos \left( { – \varphi } \right) + i\sin \left( { – \varphi } \right)} \right) \cr & – z = – r\left( {\cos \varphi + i\sin \varphi } \right) = r\left( {\cos \left( {\pi + \varphi } \right) + i\sin \left( {\pi + \varphi } \right)} \right) \cr & {1 \over z} = {z \over {\overline z .z}} = {1 \over r}\left( {\cos \varphi + i\sin \varphi } \right) \cr & k.z = kr\left( {\cos \varphi + i\sin \varphi } \right)\,\,\text{nếu}\,k > 0 \cr & kz = – kr\left( {\cos \left( {\pi + \varphi } \right) + i\sin \left( {\pi + \varphi } \right)} \right)\,\,\text{nếu}\,\,k < 0 \cr} \)
\(b)\,z = 1 + \sqrt 3 i = 2\left( {{1 \over 2} + {{\sqrt 3 } \over 2}i} \right) = 2\left( {\cos {\pi \over 3} + i\sin {\pi \over 3}} \right)\)
Áp dụng câu a) ta có: \(\overline z = 2\left( {\cos \left( { – {\pi \over 3}} \right) + i\sin \left( { – {\pi \over 3}} \right)} \right)\)
\( – z = 2\left( {\cos {{4\pi } \over 3} + i\sin {{4\pi } \over 3}} \right);\,{1 \over {\overline z }} = {1 \over 2}\left( {\cos {\pi \over 3} + i\sin {\pi \over 3}} \right)\)
\(\eqalign{ & kz = 2k\left( {\cos {\pi \over 3} + i\sin {\pi \over 3}} \right)\,\,\text{nếu}\,\,k > 0 \cr & kz = – 2k\left( {\cos {{4\pi } \over 3} + i\sin {{4\pi } \over 3}} \right)\,\text{nếu}\,\,k < 0 \cr} \)
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