Forma Polar e Exponencial de N\303\272meros ComplexosEsse \303\251 um desenvolvimento aprofundado de n\303\272meros complexos. Veremos tamb\303\251m \303\241reas de trabalho com n\303\272meros complexos e opera\303\247\303\265es com n\303\272meros complexos com a PowerTool de Algebra I & II .
[Instru\303\247\303\265es : Execute primeiramente a se\303\247\303\243o de recursos de c\303\263digos. Embora nenhum resultado seja apresentado imediatamente, essas defini\303\247\303\265es ser\303\243o usadas posteriormente nesta \303\241rea de trabalho.]
0. C\303\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 1. A Forma Polar dos N\303\272meros Complexos
Conceito de Forma Polar
Este \303\251 um n\303\272mero complexo em sua composi\303\247\303\243o padr\303\243o "parte real + parte imagin\303\241ria" .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 observamos uma representa\303\247\303\243o geom\303\251trica, podemos perceber que esse ponto forma um tri\303\242ngulo reto onde a parte real do n\303\272mero no eixo x \303\251 um cateto e a linha diagonal da origem at\303\251 o ponto \303\251 a hipotenusa. O \303\242ngulo que a hipotenusa faz com o eixo x \303\251 chamada de argumento do n\303\272mero complexo .LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpYXJndW1lbnRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiekYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdGMUYyRjItSSNtb0dGJDYtUSI7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRkFGMg==E a dist\303\242ncia da origem at\303\251 o ponto \303\251 o valor absoluto do n\303\272mero complexo.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkYWJzRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInpGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjFGMkYyLUkjbW9HRiQ2LVEiO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0ZBRjI=esses dois n\303\272meros podem identificar o ponto completamente. Por exemplo, esse mesmo n\303\272mero pode ser expresso nesta forma - que \303\251 chamada a forma trigonom\303\251trica ou polar.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 2. Convertendo Para a Forma Polar
Convertendo de forma Componente para Forma Trigonom\303\251trica
Assm como acima, o processo envolve encontrar duas coisas:
- o argumento e
- o m\303\263dulo.
Encontrando o Argumento
Podemos computar o argumento diretamente usando o Maple, mas, para fazer isso \303\240 m\303\243o, usar\303\255amos os dois catetos dos tri\303\242ngulo para encontrar a inclina\303\247\303\243o. esses dois catetos n\303\243o passam das partes imagin\303\241ria e real do n\303\272mero complexo. Uma vez que conhecemos a inclina\303\247\303\243o, sabemos que \303\251 igual \303\240 tangente do \303\242ngulo. Logo. o inverso da tangente da inclina\303\247\303\243o, \303\251 o \303\242ngulo.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Algo parece estar errado aqui.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 O que houve? Bem, o alcance do arctan \303\251 [-Pi/2, Pi/2] que s\303\243o apenas quadrantes I e IV. No entanto, este ponto est\303\241, na verdade, no quadrante II. ent\303\243o, se somarmos Pi ao nosso arctan, teremos o argumento de 2*Pi/3 . Vejamos outro exemplo.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Encontrando o Valor Absoluto
O valor absoluto de um n\303\272mero complexo \303\251 escencialmente o teorema de Pythagoras. Sabemos que os catetos de um tri\303\242ngulo s\303\243o as partes real e imagin\303\241ria de um n\303\272mero 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qcFElYXV0b0YnLUYsNiVRJGFic0YnL0ZORjhGMy1JKG1mZW5jZWRHRiQ2JC1GIzYlRkovJStleGVjdXRhYmxlR0Y4RjNGM0ZbcEZpcUYz expressando um N\303\272mero Complexo na Forma Polar
Uma vez que sabemos o argumento e o valor absoluto, \303\251 s\303\263 escrever na forma:
z = R*(cos(theta)+I*sin(theta))
Onde R = |z|, and theta = arg(z) 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 3. Convertendo da Forma Polar
Convertendo da Forma Trigonem\303\251trica para a Forma Componente
\303\211 ainda mais f\303\241cil converter da forma trigonom\303\251trica para a forma componente.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmcmVzZXRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA==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 4. A Forma Exponencial dos N\303\272meros Complexos
e a uma pot\303\252ncia imagin\303\241ria
Euler descobriu que e, quando elevado a uma pot\303\252ncia imagin\303\241ria, pode ser expresso como um n\303\272mero complexo na forma polar. Uma descoberta obviamente revolucion\303\241ria feita com a ajuda da s\303\251rie de Taylor, que voc\303\252 estudar\303\241 em c\303\241lculo.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmcmVzZXRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y9RkA=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 esses 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 valor absoluto de e^(i*theta) para qualquer valor de theta \303\251 1, e todos esses n\303\272meros se encontram no c\303\255rculo de unidades. theta pode ser qualquer n\303\272mero real.
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 e a uma Pot\303\252ncia Complexa
apesar de isso parecer ser muito mais complicado, na verdade n\303\243o \303\251 por conta das regras mais simples dos expoentes.
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Como voc\303\252 deve saber bem, a trigonometria \303\251 rica em f\303\263rmulas e identidades. Veja se lhe parecem familiares.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Essas f\303\263rmulas s\303\243o mais f\303\241ceis de se lembrar do que de se provar - usando geometria. No entanto, elas podem ser rapidamente desenvolvidas usando a forma exponencial dos n\303\272meros complexos. O que faremos \303\251 expressar uma express\303\243o exponencial em duas formas diferentes, usando as regras b\303\241sicas de expoentes. Depois expandimos cada lado para a forma polar, e equacionamos a parte real e imagin\303\241ria de cada lado.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
Voil\303\241! Voc\303\252 acaba de desenvolver duas identidades trigonom\303\251tricas pelo pre\303\247o de uma. Voc\303\252 tem uma f\303\263rmula para o seno da soma dos \303\242ngulos, e uma f\303\263rmula para o cosseno de uma soma. Quais f\303\263rmulas derivaram dessas?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Voc\303\252 pode at\303\251 impressionar os seus amigos com as suas pr\303\263prias identidades "que n\303\243o est\303\243o no 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