Question 1117670
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1.	Find the modulus and argument of the following complex numbers, and write them in trigonometric form:



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>HINT</U>.  &nbsp;&nbsp;For complex number &nbsp;z = a + bi  &nbsp;the modulus  &nbsp;&nbsp;|z| = r = {{{sqrt(a^2 + b^2)}}};  &nbsp;&nbsp;argument  t = {{{arctan(b/a)}}} &nbsp;with correction for a quadrant;  

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Trigonometric form  is  &nbsp;&nbsp;z = r*(cos(t) + i*sin(t)).



<pre>
a.	5 – 8i       Modulus  {{{sqrt(5^2+(-8)^2)}}} = {{{sqrt(89)}}}.   

                     Argument  t = {{{arctan(-8/5)}}}.

                     Trigonometric form  5 - 8i = {{{sqrt(89)*(cos(t) + i*sin(t))}}}.



b.	–1 – i       Modulus   {{{sqrt((-1)^2+(-1)^2)}}} = {{{sqrt(2)}}}.

                     Argument  t = {{{arctan((-1)/(-1)) + pi}}} = {{{pi/4 + pi}}} = {{{5pi/4}}}   (in QIII)

                     Trigonometric form  -1-i = {{{sqrt(2)*(cos(5pi/4) + i*sin(5pi/4))}}}.



c.	–5 + 12i      Modulus   {{{sqrt((-5)^2+12^2)}}} = {{{sqrt(25+144)}}} = {{{sqrt(169)}}} = 13.

                      Argument  t = {{{arctan(-12/5)+pi}}} = {{{-arctan(12/5)+pi}}}  in QII.

                      Trigonometric form  -5 + 12i = 13*(cost)+i*sin(t)).



d.	1 + {{{sqrt(3)i}}}      Modulus  {{{sqrt(1^2+(sqrt(3))^2)}}} = {{{sqrt(1+3)}}} = 2.

                      Argument  t = {{{arctan(sqrt(3))}}} = {{{pi/3}}}.

                      Trigonometric form   1 + {{{sqrt(3)*i}}} = {{{(cos(pi/3) + i*sin(pi/3))}}}.




2.	Express each of the following complex numbers in the rectangular form: a + bi.
</pre>

<pre>
a.	2 (cos30° + i sin 30°)          = {{{2*(sqrt(3)/2 + i*(1/2))}}} = {{{sqrt(3) + i}}}.


b.      &#8730;2  (cos135° + i sin 135°)      = {{{sqrt(2)*(-sqrt(2)/2 + i*(sqrt(2)/2))}}} = -1 + i.


c.	5 cis(255°)                     = {{{5*(cos(255^o) + i*sin(255^o))}}} = {{{5*cos(255^o) + 5*sin(255^o)*i}}}.


d.	 &#8730;3 cis (11&#960;/6)      = {{{sqrt(3)*(cos(11pi/6) + i*sin(11pi/6))}}} = {{{sqrt(3)*(sqrt(3)/2 + i*(-1/2))}}} = {{{3/2 - i*(1/2)}}} = 1.5 - 0.5*i.
</pre>

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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; * * * Completed and solved. * * * 

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On complex numbers, see introductory lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-taking-roots-of-complex-numbers.lesson>Solved problems on taking roots of complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-arithmetic-operations-on-complex-numbers.lesson>Solved problems on arithmetic operations on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problem-on-taking-square-roots-of-complex-numbers.lesson>Solved problem on taking square root of complex number</A>

in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Complex numbers</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.