Viết số phức sau dưới dạng lượng giác $\Large z=\dfrac{(1-i)^{10}}{(\s

Viết số phức sau dưới dạng lượng giác $\Large z=\dfrac{(1-i)^{10}}{(\sqrt{3}+i)^{9}}$

• A.

  $\Large \dfrac{1}{2^{4}}(\cos (-4 \pi)+i \cdot \sin (-4 \pi))$

• B.

  $\Large \dfrac{1}{2^{4}}(\cos (-3 \pi)+i \cdot \sin (-3 \pi))$

• C.

  $\Large \dfrac{1}{2^{4}}(\cos (2 \pi)+i \cdot \sin (2 \pi))$

• D.

  $\Large \dfrac{1}{2^{4}}(\cos (-4 \pi)-i \cdot \sin (-4 \pi))$

Ta có

$\Large \begin{array}{l}
1-i=\sqrt{2}\left(\cos \left(-\dfrac{\pi}{4}\right)+i \cdot \sin \left(-\dfrac{\pi}{4}\right)\right) 
\end{array}$

$\Large \begin{array}{l} \Rightarrow(1-i)^{10}
=\sqrt{2}^{10} \cdot\left[\cos \left(-10 \cdot \dfrac{\pi}{4}\right)+i \cdot \sin \left(-10 \cdot \dfrac{\pi}{4}\right)\right] \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=2^{5}\left[\cos \left( \dfrac{5 \pi}{2}\right)+i \cdot \sin \left(-\dfrac{5 \pi}{2}\right)\right]
\end{array}$

Lại có

$\Large \begin{array}{l}
\sqrt{3}+i=2\left(\cos \dfrac{\pi}{6}+i \cdot \sin \dfrac{\pi}{6}\right) \\
\Rightarrow(\sqrt{3}+i)^{9}=2^{9} \cdot\left(\cos 9 \cdot \dfrac{\pi}{6}+i \cdot \sin 9 \cdot \dfrac{\pi}{6}\right)
\end{array}$

$\Large =2^{9} \cdot\left(\cos \dfrac{3 \pi}{2}+i \cdot \sin \dfrac{3 \pi}{2}\right)$

Do đó

$\Large \begin{array}{l}
z=\dfrac{(1-i)^{10}}{(\sqrt{3}+i)^{9}} \\
=\dfrac{2^{5} \cdot\left[\cos \left(\dfrac{-5 \pi}{2}\right)+i \cdot \sin \left(\dfrac{-5 \pi}{2}\right)\right]}{2^{9} \cdot\left(\cos \dfrac{3 \pi}{2}+i \cdot \sin \dfrac{3 \pi}{2}\right)} \\
=\dfrac{1}{2^{4}}(\cos (-4 \pi)+i \cdot \sin (-4 \pi))
\end{array}$

Chọn A