Question

I will Thank anyone who can answer this for me!!

Answer 1

$\large\begin{array}{l}\\\\ \textsf{This is a complex number in }\mathsf{a+bi}\textsf{ form:}\\\\ \mathsf{z=5-3i}\\\\ \textsf{where }\mathsf{a=5~~and~~b=-3.} \end{array}$


$\large\begin{array}{l}\\\\ \bullet~~\textsf{1}\mathsf{^{st}}\textsf{ step: Find the absolute value of z (complex modulus of z):}\\\\ \mathsf{|z|=\sqrt{a^2+b^2}}\\\\ \mathsf{|z|=\sqrt{5^2+(-3)^2}}\\\\ \mathsf{|z|=\sqrt{25+9}}\\\\ \mathsf{|z|=\sqrt{34}} \end{array}$


$\large\begin{array}{l}\bullet~~\textsf{2}\mathsf{^{nd}}\textsf{ step: Find the argument }\mathsf{\theta}\textsf{ of z:}\\\\ \mathsf{arg(z)=\theta}\\\\ \textsf{is an angle that satisfies the following conditions:}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{a}{|z|}=\dfrac{a}{\sqrt{a^2+b^2}}}\\\\ \mathsf{sin\,\theta=\dfrac{b}{|z|}=\dfrac{b}{\sqrt{a^2+b^2}}} \end{cases}\qquad\quad \mathsf{0\le \theta$


$\large\begin{array}{l} \textsf{For }\mathsf{z=5-3i,}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{5}{\sqrt{34}}}\\\\ \mathsf{sin\,\theta=\dfrac{-3}{\sqrt{34}}} \end{cases}\\\\\\ \textsf{Since}\\\\ \mathsf{cos\,\theta>0~~and~~sin\,\theta$


$\large\begin{array}{l} \mathsf{tan\,\theta=\dfrac{b}{a}}\\\\ \mathsf{tan\,\theta=\dfrac{-3}{5}}\\\\ \mathsf{tan\,\theta=-\,\dfrac{3}{5}}\\\\ \mathsf{\theta=2\pi+tan^{-1}\!\left(-\,\dfrac{3}{5}\right)}\\\\ \mathsf{\theta=2\pi-tan^{-1}\!\left(\dfrac{3}{5}\right)}\qquad\quad\textsf{(exact form)}\\\\ \mathsf{\theta\approx 6.283-0.540}\\\\ \mathsf{\theta\approx 5.743~rad~~(329.04^\circ)} \end{array}$


$\large\begin{array}{l}\bullet~~\textsf{3}\mathsf{^{rd}}\textsf{ step: Express z in the trigonometric form:}\\\\ \begin{array}{rcl} \mathsf{z}&\!\!=\!\!&\mathsf{|z|\cdot (cos\,\theta+i\cdot sin\,\theta)}\\\\ \mathsf{5-3i}&\!\!\approx \!\!&\mathsf{\sqrt{34}\cdot \big[cos(5.743)+i\cdot sin(5.743)\big]\qquad\quad\checkmark} \end{array} \end{array}$


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


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