document.write( "Question 1117670: 1. Find the modulus and argument of the following complex numbers, and write them in trigonometric form:
\n" ); document.write( "a. 5 – 8i
\n" ); document.write( "b. –1 – i
\n" ); document.write( "c. –5 + 12i
\n" ); document.write( "d. 1 + √3i \r
\n" ); document.write( "\n" ); document.write( "2. Express each of the following complex numbers in the rectangular form: a + bi.\r
\n" ); document.write( "\n" ); document.write( "a. 2 (cos30° + i sin 30°)
\n" ); document.write( "b. √2 (cos135° + i sin 135°)
\n" ); document.write( "c. 5 cis(255°)
\n" ); document.write( "d. √3 cis (11π/6)
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #732793 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "1. Find the modulus and argument of the following complex numbers, and write them in trigonometric form:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            HINT.   For complex number  z = a + bi  the modulus   |z| = r = \"sqrt%28a%5E2+%2B+b%5E2%29\";   argument t = \"arctan%28b%2Fa%29\"  with correction for a quadrant; \r
\n" ); document.write( "\n" ); document.write( "                          Trigonometric form is   z = r*(cos(t) + i*sin(t)).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "a.	5 – 8i       Modulus  \"sqrt%285%5E2%2B%28-8%29%5E2%29\" = \"sqrt%2889%29\".   \r\n" );
document.write( "\r\n" );
document.write( "                     Argument  t = \"arctan%28-8%2F5%29\".\r\n" );
document.write( "\r\n" );
document.write( "                     Trigonometric form  5 - 8i = \"sqrt%2889%29%2A%28cos%28t%29+%2B+i%2Asin%28t%29%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "b.	–1 – i       Modulus   \"sqrt%28%28-1%29%5E2%2B%28-1%29%5E2%29\" = \"sqrt%282%29\".\r\n" );
document.write( "\r\n" );
document.write( "                     Argument  t = \"arctan%28%28-1%29%2F%28-1%29%29+%2B+pi\" = \"pi%2F4+%2B+pi\" = \"5pi%2F4\"   (in QIII)\r\n" );
document.write( "\r\n" );
document.write( "                     Trigonometric form  -1-i = \"sqrt%282%29%2A%28cos%285pi%2F4%29+%2B+i%2Asin%285pi%2F4%29%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "c.	–5 + 12i      Modulus   \"sqrt%28%28-5%29%5E2%2B12%5E2%29\" = \"sqrt%2825%2B144%29\" = \"sqrt%28169%29\" = 13.\r\n" );
document.write( "\r\n" );
document.write( "                      Argument  t = \"arctan%28-12%2F5%29%2Bpi\" = \"-arctan%2812%2F5%29%2Bpi\"  in QII.\r\n" );
document.write( "\r\n" );
document.write( "                      Trigonometric form  -5 + 12i = 13*(cost)+i*sin(t)).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "d.	1 + \"sqrt%283%29i\"      Modulus  \"sqrt%281%5E2%2B%28sqrt%283%29%29%5E2%29\" = \"sqrt%281%2B3%29\" = 2.\r\n" );
document.write( "\r\n" );
document.write( "                      Argument  t = \"arctan%28sqrt%283%29%29\" = \"pi%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "                      Trigonometric form   1 + \"sqrt%283%29%2Ai\" = \"%28cos%28pi%2F3%29+%2B+i%2Asin%28pi%2F3%29%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.	Express each of the following complex numbers in the rectangular form: a + bi.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "a.	2 (cos30° + i sin 30°)          = \"2%2A%28sqrt%283%29%2F2+%2B+i%2A%281%2F2%29%29\" = \"sqrt%283%29+%2B+i\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "b.      √2  (cos135° + i sin 135°)      = \"sqrt%282%29%2A%28-sqrt%282%29%2F2+%2B+i%2A%28sqrt%282%29%2F2%29%29\" = -1 + i.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "c.	5 cis(255°)                     = \"5%2A%28cos%28255%5Eo%29+%2B+i%2Asin%28255%5Eo%29%29\" = \"5%2Acos%28255%5Eo%29+%2B+5%2Asin%28255%5Eo%29%2Ai\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "d.	 √3 cis (11π/6)      = \"sqrt%283%29%2A%28cos%2811pi%2F6%29+%2B+i%2Asin%2811pi%2F6%29%29\" = \"sqrt%283%29%2A%28sqrt%283%29%2F2+%2B+i%2A%28-1%2F2%29%29\" = \"3%2F2+-+i%2A%281%2F2%29\" = 1.5 - 0.5*i.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "=====================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                    =====================\r
\n" ); document.write( "\n" ); document.write( "             * * * Completed and solved. * * * \r
\n" ); document.write( "\n" ); document.write( "                    =====================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On complex numbers, see introductory lessons\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Complex numbers\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );