Giải bài 27 trang 205 SGK giải tích nâng cao 12

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 $\left( k<0 \right)$