Podeli

812.
Trigonometrijski i eksponencijalni oblik

TEKST ZADATKA

Izračunati:

\frac{(-1+i\sqrt3)^{15}}{(1-i)^{20}}+\frac{(-1-i\sqrt3)^{15}}{(1+i)^{20}}

REŠENJE ZADATKA

Zapisati brojeve u trigonometrijskom obliku : $z=|z|\cdot(\cos{\varphi}+i\sin{\varphi}).$

-1+i\sqrt3=2\big(\cos\frac{2\pi}3+i\sin\frac{2\pi}3\big) \\ 1-i=\sqrt2\big(\cos(-\frac{\pi}4)+i\sin(-\frac{\pi}4)\big) \\ -1-i\sqrt3=2\big(\cos(-\frac{2\pi}3)+i\sin(-\frac{2\pi}3)\big) \\ 1+i=\sqrt{2}\big(\cos{\frac{\pi}{4}}+i\sin{\frac{\pi}{4}}\big)

Uvrstiti trigonometrijske oblike kompleksnih brojeva u početni izraz.

\frac{(2\big(\cos\frac{2\pi}3+i\sin\frac{2\pi}3\big))^{15}}{(\sqrt2\big(\cos(-\frac{\pi}4)+i\sin(-\frac{\pi}4)\big))^{20}}+\frac{(2\big(\cos(-\frac{2\pi}3)+i\sin(-\frac{2\pi}3)\big))^{15}}{(\sqrt{2}(\cos{\frac{\pi}{4}}+i\sin{\frac{\pi}{4}}))^{20}}

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

\frac{2^{15}\big(\cos(15\cdot\frac{2\pi}3)+i\sin(15\cdot\frac{2\pi}3)\big)}{(\sqrt2)^{20}\big(\cos(-20\cdot\frac{\pi}4)+i\sin(-20\cdot\frac{\pi}4)\big)}+\frac{2^{15}\big(\cos(-15\cdot\frac{2\pi}3)+i\sin(-15\cdot\frac{2\pi}3)\big)}{(\sqrt{2})^{20}(\cos(20\cdot\frac{\pi}{4})+i\sin(20\cdot\frac{\pi}{4}))} \\ \frac{2^{15}\big(\cos10\pi+i\sin10\pi\big)}{2^{10}\big(\cos(-5\pi)+i\sin(-5\pi)\big)}+\frac{2^{15}\big(\cos(-10\pi)+i\sin(-10\pi)\big)}{2^{10}(\cos5\pi+i\sin5\pi)}

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

2^5 \cdot(\cos(10\pi-(-5\pi))+i\sin(10\pi-(-5\pi))) +2^5\cdot(\cos(-10\pi-5\pi)+i\sin(-10\pi-5\pi))\\ 2^5 (\cos15\pi+i\sin15\pi) +2^5(\cos(-15\pi)+i\sin(-15\pi)

Svesti uglove na prvi kvadrant.

2^5 (\cos{\pi}+i\sin{\pi}) +2^5(\cos{\pi}-i\sin{\pi})

DODATNO OBJAŠNJENJE

\cos{(15\pi)}=\cos{(14\pi+\pi)}=\cos{(\pi)}\\ \sin{(15\pi)}=\sin{(14\pi+\pi)}=\sin{(\pi)}\\ \cos{(-15\pi)}=\cos{(15\pi)}=\cos{(\pi)}\\ \sin{(15\pi)}=-\sin{(15\pi)}=-\sin{(\pi)}\\

Zapisati kompleksne brojeve u algebarskom obliku.

2^5(-1+i\cdot 0) + 2^5(-1-i\cdot 0)\\ -2^5-2^5\\ -32-32 \\ -64

812.Trigonometrijski i eksponencijalni oblik