Giải bài 27 trang 205 SGK giải tích nâng cao 12
Hãy tìm dạng lượng giác của các số phức \(\overline{z};-z;\dfrac{1}{z};kz\,\left( k\in {{\mathbb{R}}^{*}} \right)\) trong mỗi trường hợp sau:
a) \(z=r\left( \cos \varphi +i\sin \varphi \right)\,\left( r>0 \right) \);
b) \(z=1+\sqrt{3}i \);
a) \(z=r\left( \cos \varphi +i\sin \varphi \right)\,\left( r>0 \right) \)
Suy ra
\(\begin{aligned} \overline{z}&=r\left( \cos \varphi -i\sin \varphi \right) \\ & =r\left[ \cos \left( -\varphi \right)+i\sin \left( -\varphi \right) \right] \\ \end{aligned} \)
\(\begin{aligned} -z&=-r\left( \cos \varphi +i\sin \varphi \right) \\ & =r\left( -\cos \varphi -i\sin \varphi \right) \\ & =r\left[ \cos \left( \varphi +\pi \right)+i\sin \left( \varphi +\pi \right) \right] \\ \end{aligned} \)
\(\dfrac{1}{\overline{z}}=\dfrac{z}{\overline{z}.z}\)
\(\begin{aligned} \overline{z}.z&=r\left[ \cos \left( -\varphi \right)+i\sin \left( -\varphi \right) \right].r\left( \cos \varphi +i\sin \varphi \right) \\ & ={{r}^{2}}\left[ \cos \left( -\varphi +\varphi \right)+i\sin \left( -\varphi +\varphi \right) \right] \\ & ={{r}^{2}} \\ \end{aligned} \)
\(\Rightarrow \dfrac{z}{\overline{z}.z}=\dfrac{1}{r}\left( \cos \varphi +i\sin \varphi \right)\)
\(kz=kr\left( \cos \varphi +i\sin \varphi \right)\) nếu \(\left( k>0 \right) \)
\(kz=-kr\left( \cos \varphi +i\sin \varphi \right)\) nếu \(\left( k<0 \right) \).
b) Ta có \(z=1+\sqrt{3}i=2\left( \dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \right)=2\left( \cos \dfrac{\pi }{3}+i\sin \dfrac{\pi }{3} \right)\)
Suy ra
\(\overline{z}=2\left[ \cos \left( -\dfrac{\pi }{3} \right)+i\sin \left( -\dfrac{\pi }{3} \right) \right]\)
\(\begin{aligned} -z&=2\left[ \cos \left( \pi +\dfrac{\pi }{3} \right)+i\sin \left( \pi +\dfrac{\pi }{3} \right) \right] \\ & =2\left( \cos \dfrac{4\pi }{3}+i\sin \dfrac{4\pi }{3} \right) \\ \end{aligned} \)
\(\dfrac{1}{\overline{z}}=\dfrac{1}{2}\left( \cos \dfrac{\pi }{3}+i\sin \dfrac{\pi }{3} \right)\)
\(kz=2k\left( \cos \dfrac{\pi }{3}+i\sin \dfrac{\pi }{3} \right)\,\,\left( k>0 \right) \\ kz=-2k\left( \cos \dfrac{\pi }{3}+i\sin \dfrac{\pi }{3} \right)\,\left( k<0 \right) \)
Ghi nhớ: Cho \(z=r\left( \cos \varphi +i\sin \varphi \right)\,\left( r>0 \right) \)
Khi đó
\(\begin{aligned} \overline{z}=r\left[ \cos \left( -\varphi \right)+i\sin \left( -\varphi \right) \right] \\ \end{aligned} \)
\(\begin{aligned} -z=r\left[ \cos \left( \varphi +\pi \right)+i\sin \left( \varphi +\pi \right) \right] \\ \end{aligned} \)
\(\dfrac{1}{\overline{z}}=\dfrac{1}{r}\left( \cos \varphi +i\sin \varphi \right)\)
\(kz=kr\left( \cos \varphi +i\sin \varphi \right)\) nếu \(\left( k>0 \right) \)
\(kz=-kr\left( \cos \varphi +i\sin \varphi \right)\) nếu \(\left( k<0 \right) \)