Stepen količnika: ((1 + i√3)/(1 - i)) na 20

Odrediti trigonometrijski oblik kompleksnog broja $1+i\sqrt3$ po formuli: $z=|z|\cdot(\cos{(\text{arg}(z))}+i\sin{(\text{arg}(z))}).$

1+i\sqrt3=2\big(\cos\frac{\pi}3+i\sin\frac{\pi}3\big)

Odrediti trigonometrijski oblik kompleksnog broja $1-i$ po formuli: $z=|z|\cdot(\cos{(\text{arg}(z))}+i\sin{(\text{arg}(z))}).$

1-i=\sqrt2\big(\cos(-\frac{\pi}4)+i\sin(-\frac{\pi}4)\big)

Uvrstiti trigonometrijske oblike kompleksnih brojeva u početni izraz.

\bigg(\frac{2\big(\cos\frac{\pi}3+i\sin\frac{\pi}3\big)}{\sqrt2\big(\cos(-\frac{\pi}4)+i\sin(-\frac{\pi}4)\big)}\bigg)^{20} \\ \frac{\big(2\big(\cos\frac{\pi}3+i\sin\frac{\pi}3\big)\big)^{20}}{\big(\sqrt2\big(\cos(-\frac{\pi}4)+i\sin(-\frac{\pi}4)\big)\big)^{20}}

Primeniti formulu za izračunavanje stepena kompleksnog broja: $\big(r(\cos{\alpha}+i\sin{\alpha})\big)^n=r^n(\cos(n\cdot\alpha)+i\sin(n\cdot\alpha))$

\frac{2^{20}\big(\cos(20\cdot\frac{\pi}3)+i\sin(20\cdot\frac{\pi}3)\big)}{(\sqrt2)^{20}\big(\cos(-20\cdot\frac{\pi}4)+i\sin(-20\cdot\frac{\pi}4)\big)} \\ \frac{2^{20}\big(\cos(\frac{20\pi}3)+i\sin(\frac{20\pi}3)\big)}{2^{10}\big(\cos(-5\pi)+i\sin(-5\pi)\big)}

Svesti uglove na prvi kvadrant.

\frac{2^{20}\big(\cos(\frac{2\pi}3)+i\sin(\frac{2\pi}3)\big)}{2^{10}\big(\cos\pi+i\sin\pi\big)}

Primeniti formulu za količnik kompleksnog broja: $\frac{z_1}{z_2}=\frac{|z_1|}{|z_2|}(\cos(\varphi_1-\varphi_2) + i\sin(\varphi_1-\varphi_2)).$

\frac{2^{20}}{2^{10}}\big(\cos(\frac{2\pi}3-\pi)+i\sin(\frac{2\pi}3-\pi)\big) \\ 2^{10}\big(\cos(-\frac{\pi}3)+i\sin(-\frac{\pi}3)\big)

Zapisati kompleksni broj u algebarskom obliku.

2^{10}\big(\frac12+i\cdot(-\frac{\sqrt3}2)\big) \\ 2^9\big(1-i\cdot\sqrt3\big) \\ 512-512\sqrt3i

$\big(\frac{1+i\sqrt3}{1-i}\big)^{20}$