Trigonometrijski oblik kompleksnog broja

Ponekad je kompleksne brojeve potrebno pisati u trigonometrijskom obliku

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7a8683d333ee55e56137e3a81882de44a06861" alt="{displaystyle a+bi=rho (cos phi +isin phi ),}" width="11155.1" height="1223.9">$ ,

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d15e943d2803b86b2f3fe74cfa4d3a8e83c0d41" alt="{displaystyle rho ={sqrt {a^{2}+b^{2}}}, phi =arctan {frac {b}{a}}}" width="12407.7" height="2300.3">$ , za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" alt="{displaystyle a>0}" width="2364.1" height="936.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667bb2b83357c87ef01d5a915c782d94f84c52e7" alt="{displaystyle phi =pi +arctan {frac {b}{a}}}" width="7567.2" height="2300.3">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c" alt="{displaystyle a<0}" width="2364.1" height="936.9">$ ; kada je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" alt="{displaystyle a=0}" width="2364.1" height="936.9">$ onda je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a633e700dd0b05a98f93102f35421b3ccbd9fa" alt="{displaystyle phi ={frac {pi }{2}}}" width="2864.1" height="2013.3">$ , ako je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" alt="{displaystyle b>0}" width="2264.1" height="936.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c327eced4f372f8635ce9d0cf33a2e9d44c3a4" alt="{displaystyle phi =-{frac {pi }{2}}}" width="3642.6" height="2013.3">$ , ako je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1319de18afa795d6489f49303614c84472f6d1ed" alt="{displaystyle b<0}" width="2264.1" height="936.9">$ .

Broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" alt="{displaystyle rho }" width="517.5" height="936.9">$ se naziva modul kompleksnog broja , а $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" alt="{displaystyle phi }" width="596.5" height="1080.4">$ је argument kompleksnog broja

Množiti kompleksne brojeve pogodno je u ovom obliku .

Moavrova formula

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c39bcdb3049921f9675a2d3359028a70df03" alt="{displaystyle (cos phi +isin phi )^{n}=cos nphi +isin nphi ,}" width="15666.2" height="1223.9">$ .

Često se predstavljaju vektorima u kompleksnoj ravni .

Dužina vektora $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" alt="{displaystyle rho }" width="517.5" height="936.9">$ је modul kompleksnog broja.

Izvor: Wikipedia

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Tag Archives: trigonometrijski oblik kompleksnog broja

Kako izgleda trigonometrijski oblik kompleksnog broja?

Trigonometrijski oblik kompleksnog broja

Moavrova formula

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c39bcdb3049921f9675a2d3359028a70df03" alt="{displaystyle (cos phi +isin phi )^{n}=cos nphi +isin nphi ,}" width="15666.2" height="1223.9">$ .

Stranica o prirodi i svemu vezanom za prirodu.

Trigonometrijski oblik kompleksnog broja

Moavrova formula

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c39bcdb3049921f9675a2d3359028a70df03" alt="{displaystyle (cos phi +isin phi )^{n}=cos nphi +isin nphi ,}" width="15666.2" height="1223.9"> .

Stranica o prirodi i svemu vezanom za prirodu.

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c39bcdb3049921f9675a2d3359028a70df03" alt="{displaystyle (cos phi +isin phi )^{n}=cos nphi +isin nphi ,}" width="15666.2" height="1223.9">$ .