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CLASE 16

CLASE 16. POTENCIACIÓN DE NÚMEROS COMPLEJOS EN FORMA TRIGONOMÉTRICA. Recuerda que:. Los números complejos en forma trigonométrica se expresan. z=  (cos  + i sen  ). z=  cis . :.  . cos (  +  ) + i sen (  +  ). cos (  –  ) + i sen (  –  ). Recuerda que:.

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CLASE 16

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  1. CLASE 16

  2. POTENCIACIÓN DE NÚMEROS COMPLEJOS EN FORMA TRIGONOMÉTRICA

  3. Recuerda que: Los números complejos en forma trigonométrica se expresan z= (cos+i sen) z= cis :

  4.  cos(+)+i sen(+) cos(–)+i sen(–) Recuerda que: z=(cos+i sen) y=(cos+isen) Multiplicación . Z .Y= . División Z:Y=

  5. Efectúa: i 2 1 2 i i = (0,9)2+ 2 . 0,9 . + 2 1 2 1 2 0,9 + Ejemplo 1 = 0,81+ 0,9 . i + 0,25(–1) –0,25 = 0,56+ 0,9 i .

  6. Si z = (cos+i sen) es un número complejo y n, se cumple: zn = n(cos n+i sen n) Teorema de Moivre :

  7. Notación abreviada zn= n(cos n+i sen n) zn= nci s n .

  8. Ejemplo Calcula znsi: z= 0,2(cos 23o+ i sen23o) n = 3 z3= (0,2)3(cos 3.23o + i sen 3.23o) z3= (0,008)(cos 69o + i sen 69o) z3= 0,0029+ 0,0075 i .

  9. z5 = 1282 cis 315o ESTUDIO INDIVIDUAL Calcula zn si: a) z = 3 cis 26o n= 6 z6 = 729 cis 156o . b) z = –2 + 2i n= 5

  10. Calcula znsi: =12+(–1)2 =2 : z = 1 – i n = 4 a = 1 b = – 1 tan  = – 1 tan  = 1 =45o Afijo (1;–1)  =315o Cuarto cuadrante: 360o–= 360o–45o=315o

  11. z4= (2)4(cos 4·315o + i sen 4·315o) z4= (2)2(cos 12600 + isen 12600) z4= 4 (cos 1800 + isen 1800) z4= 4 (–1 + 0) z4= –4 .

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