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forsale [732]
3 years ago
13

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

Mathematics
1 answer:
Licemer1 [7]3 years ago
5 0
\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}


If you're having problems understanding this answer, try seeing it through your browser: brainly.com/question/2159691


\large\textsf{I hope it helps.}


Tags: <em>complex number trigonometric form trig polar absolute value modulus argument angle</em>

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