812.

(−1+i3)15(1−i)20+(−1−i3)15(1+i)20\frac{(-1+i\sqrt3)^{15}}{(1-i)^{20}}+\frac{(-1-i\sqrt3)^{15}}{(1+i)^{20}}

TEKST ZADATKA

Izračunati:

(−1+i3)15(1−i)20+(−1−i3)15(1+i)20\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∣⋅(cos⁡φ+isin⁡φ).z=|z|\cdot(\cos{\varphi}+i\sin{\varphi}).

−1+i3=2(cos⁡2π3+isin⁡2π3)1−i=2(cos⁡(−π4)+isin⁡(−π4))−1−i3=2(cos⁡(−2π3)+isin⁡(−2π3))1+i=2(cos⁡π4+isin⁡π4)-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.

(2(cos⁡2π3+isin⁡2π3))15(2(cos⁡(−π4)+isin⁡(−π4)))20+(2(cos⁡(−2π3)+isin⁡(−2π3)))15(2(cos⁡π4+isin⁡π4))20\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: (r(cos⁡α+isin⁡α))n=rn(cos⁡(n⋅α)+isin⁡(n⋅α))\big(r(\cos{\alpha}+i\sin{\alpha})\big)^n=r^n(\cos(n\cdot\alpha)+i\sin(n\cdot\alpha))

215(cos⁡(15⋅2π3)+isin⁡(15⋅2π3))(2)20(cos⁡(−20⋅π4)+isin⁡(−20⋅π4))+215(cos⁡(−15⋅2π3)+isin⁡(−15⋅2π3))(2)20(cos⁡(20⋅π4)+isin⁡(20⋅π4))215(cos⁡10π+isin⁡10π)210(cos⁡(−5π)+isin⁡(−5π))+215(cos⁡(−10π)+isin⁡(−10π))210(cos⁡5π+isin⁡5π)\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: z1z2=∣z1∣∣z2∣(cos⁡(φ1−φ2)+isin⁡(φ1−φ2)).\frac{z_1}{z_2}=\frac{|z_1|}{|z_2|}(\cos(\varphi_1-\varphi_2) + i\sin(\varphi_1-\varphi_2)).

25⋅(cos⁡(10π−(−5π))+isin⁡(10π−(−5π)))+25⋅(cos⁡(−10π−5π)+isin⁡(−10π−5π))25(cos⁡15π+isin⁡15π)+25(cos⁡(−15π)+isin⁡(−15π)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.

25(cos⁡π+isin⁡π)+25(cos⁡π−isin⁡π)2^5 (\cos{\pi}+i\sin{\pi}) +2^5(\cos{\pi}-i\sin{\pi})
DODATNO OBJAŠNJENJE

Zapisati kompleksne brojeve u algebarskom obliku.

25(−1+i⋅0)+25(−1−i⋅0)−25−25−32−32−642^5(-1+i\cdot 0) + 2^5(-1-i\cdot 0)\\ -2^5-2^5\\ -32-32 \\ -64