Mathematics: Find the trigonometric form of the complex number -3+2i

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Find the trigonometric form of the complex number

Ответ:

mariaveliz2201

10.06.2020

$\large\begin{array}{l}\\\\ \textsf{This is a complex number in 
}\mathsf{a+bi}\textsf{ form:}\\\\ \mathsf{z=-3+2i}\\\\ 
\textsf{where }\mathsf{a=-3~~and~~b=2.} \end{array}$

$\large\begin{array}{l}\\\\
 \bullet~~\textsf{1}\mathsf{^{st}}\textsf{ step: Find the absolute value
 of z (complex modulus of z):}\\\\ \mathsf{|z|=\sqrt{a^2+b^2}}\\\\ 
\mathsf{|z|=\sqrt{(-3)^2+2^2}}\\\\ 
\mathsf{|z|=\sqrt{9+4}}\\\\ \mathsf{|z|=\sqrt{13}} \end{array}$

$\large\begin{array}{l}\bullet~~\textsf{2}\mathsf{^{nd}}\textsf{
 step: Find the argument }\mathsf{\theta}\textsf{ of z:}\\\\ 
\mathsf{arg(z)=\theta}\\\\ \textsf{is an angle that satisfies the 
following conditions:}\\\\ \begin{cases} 
\mathsf{cos\,\theta=\dfrac{a}{|z|}=\dfrac{a}{\sqrt{a^2+b^2}}}\\\\ 
\mathsf{sin\,\theta=\dfrac{b}{|z|}=\dfrac{b}{\sqrt{a^2+b^2}}} 
\end{cases}\qquad\quad-\pi<\theta \le \pi \end{array}$

$\large\begin{array}{l} \textsf{For }\mathsf{z=-3+2i,}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{-3}{\sqrt{13}}}\\\\ \mathsf{sin\,\theta=\dfrac{2}{\sqrt{13}}} \end{cases}\\\\\\ \textsf{Since}\\\\ \mathsf{cos\,\theta<0~~and~~sin\,\theta0,}\\\\ \theta\textsf{ lies in the 2}\mathsf{^{nd}}\textsf{ quadrant.} \end{array}$

$\large\begin{array}{l} \mathsf{tan\,\theta=\dfrac{b}{a}}\\\\ \mathsf{tan\,\theta=\dfrac{2}{-3}}\\\\ \mathsf{tan\,\theta=-\,\dfrac{2}{3}}\\\\ \mathsf{\theta=\pi+tan^{-1}\!\left(-\,\dfrac{2}{3}\right)}\\\\ \mathsf{\theta=\pi-tan^{-1}\!\left(\dfrac{2}{3}\right)}\qquad\quad\textsf{(exact form)}\\\\ \mathsf{\theta\approx 3.142-0.588}\\\\ \mathsf{\theta\approx 2.554~rad~~(146.31^\circ)} \end{array}$

$\large\begin{array}{l}\bullet~~\textsf{3}\mathsf{^{rd}}\textsf{ step: Express z in the trigonometric form:}\\\\ \begin{array}{rcl} \mathsf{z}&\!\!=\!\!&\mathsf{|z|\cdot (cos\,\theta+i\cdot sin\,\theta)}\\\\ \mathsf{-3+2i}&\!\!\approx \!\!&\mathsf{\sqrt{13}\cdot \big[cos(146.31^\circ)+i\cdot sin(146.31^\circ)\big]\qquad\quad\checkmark} \end{array} \end{array}$

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$\large\textsf{I hope this helps.}$

Tags: complex number trigonometric form trig absolute value modulus argument angle

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Find the trigonometric form of the complex number -3+2i

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