Question

Escreva o complexo na forma trigonométrica: z = -7 - 7i

Answer 1

z = - 7 - 7i
a = - 7, b = - 7
Módulo de z:
IzI = √(a² + b²)
IzI = √[ (- 7)² + (- 7)²] = √(49 + 49) = 7√2
cos (Ф) =  a 
               IzI
cos(Ф) =  - 7    = -  √2               
                7√2         2
sen(Ф) =  b 
              IzI
sen(Ф) =  - 7  = -  √2               
                7√2        2
A forma trigonométrica de z é:
z = IzI(cos(Ф) + sen(Ф)i)
z = 7√2[cos(225°) + sen(225°)i]

Answer 2

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módulo de um número complexo

$\Large\boxed{\begin{array}{l}\sf Seja~z=a+bi~um~n\acute umero~complexo.\\\sf o~m\acute odulo~de~z~\acute e~dado~por:\\\Huge\boxed{\boxed{\boxed{\boxed{\sf \rho=\sqrt{a^2+b^2}}}}}\end{array}}$

Argumento de um número complexo

$\Large\boxed{\begin{array}{l}\sf \acute e~o~\hat angulo~\theta~tal~que\\\boxed{\boxed{\sf cos(\theta)=\dfrac{a}{\rho}}}~e~\boxed{\boxed{\sf sen(\theta)=\dfrac{b}{\rho}}}\end{array}}$

Forma trigonométrica de  um número complexo

$\Large\boxed{\begin{array}{l}\sf \acute e~a~representac_{\!\!,}\tilde ao~do~complexo~z=a+bi~\\\sf em~func_{\!\!,}\tilde ao~de~seu~m\acute odulo~e~de~seu~argumento.\\\sf a~forma~trigonom\acute etrica~de~z~\acute e:\\\huge\boxed{\boxed{\boxed{\boxed{\sf z=\rho[cos(\theta)+i~sen(\theta)]}}}}\end{array}}$

$\sf z=-7-7i\\\sf \rho=\sqrt{(-7)^2+(-7)^2}\\\sf\rho=\sqrt{49+49}\\\sf \rho=\sqrt{49\cdot2}\\\sf\rho=7\sqrt{2}\\\begin{cases}\sf cos(\theta)=\dfrac{-\diagup\!\!\!7}{\diagup\!\!\!7\sqrt{2}}=-\dfrac{\sqrt{2}}{2}\\\sf sen(\theta)=\dfrac{-\diagup\!\!\!7}{\diagup\!\!\!7\sqrt{2}}=-\dfrac{\sqrt{2}}{2}\end{cases}\implies\tt \theta=\dfrac{5\pi}{4}\\\huge\boxed{\boxed{\boxed{\boxed{\sf z=7\sqrt{2}\bigg[cos\bigg(\dfrac{5\pi}{4}\bigg)+i~sen\bigg(\dfrac{5\pi}{4}\bigg)\bigg]}}}}$