document.write( "Question 1071888: Convert to trigonometric form: \"+z+=+4+\" square root of 3 space plus space 4 i\r
\n" ); document.write( "\n" ); document.write( "A.) 8(cos60 + i sin60)
\n" ); document.write( "B.) 8(cos30 + i sin30)
\n" ); document.write( "C.) 8(sin30 + i cos30)
\n" ); document.write( "D.) 8(sin60 + i cos60) \n" ); document.write( "
Algebra.Com's Answer #686827 by KMST(5328)\"\" \"About 
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\"4sqrt%283%29%2B4i=8%28sqrt%283%29%2F2%2BI%281%2F2%29%29\"
\n" ); document.write( "\"cos%2830%5Eo%29=sqrt%283%29%2F2\" and \"sin%2830%5Eo%29=1%2F2\" , so
\n" ); document.write( "\"4sqrt%283%29%2B4i=8%28cos%2830%5Eo%29%2Bi%2Asin%2830%5Eo%29%29\"
\n" ); document.write( "
\n" ); document.write( "TIP:
\n" ); document.write( "If they give you a complex number with
\n" ); document.write( "a non-zero real part, and
\n" ); document.write( "a different non-zero imaginary part,
\n" ); document.write( "and the angle has to be a nice round number,
\n" ); document.write( "one part will be \"sqrt%283%29\" or \"-sqrt%283%29\"
\n" ); document.write( "times the other part,
\n" ); document.write( "Because the only exact angles with
\n" ); document.write( "non-zero, different, and simple-to-write
\n" ); document.write( "cosine and sine are \"30%5Eo\" , \"60%5Eo\" in quadrant 1,
\n" ); document.write( "and angles symmetrical to those in the other quadrants.
\n" ); document.write( "\"cos%2830%5Eo%29=sqrt%283%29%2F2=sin%2860%5Eo%29\" and
\n" ); document.write( "\"cos%2860%5Eo%29=1%2F2=sin%2830%5Eo%29\" .
\n" ); document.write( "If both parts are positive, it is one of those angles.
\n" ); document.write( "If there are negative parts,
\n" ); document.write( "it is a symmetrical angle on another quadrant.
\n" ); document.write( "For example, \"-1%2Bsqrt%283%29%2Ai\"
\n" ); document.write( "has a negative real part, and a positive imaginary part,
\n" ); document.write( "so it would be plotted in quadrant 2.
\n" ); document.write( "The real part has to be cosine times something,
\n" ); document.write( "so it is \"%281%2F2%29%28-2%29=-1\"
\n" ); document.write( "The quadrant 1 angle with \"1%2F2\" for a cosine
\n" ); document.write( "is \"60%5Eo\" .
\n" ); document.write( "To get the symmetrical angle from quadrant 2,
\n" ); document.write( "You subtract from \"180%5Eo\" ,
\n" ); document.write( "to get \"180%5Eo-60%5Eo=120%5Eo\" in quadrant 2.
\n" ); document.write( "So,
\n" ); document.write( " .
\n" ); document.write( "If both parts are negative,
\n" ); document.write( "the complex number would be plotted in quadrant 3,
\n" ); document.write( "where the angle would be \"180%5Eo\" plus
\n" ); document.write( "the symmetrical reference angle from quadrant 1.
\n" ); document.write( "If the real part is positive,
\n" ); document.write( "and the imaginary part negative,
\n" ); document.write( "You are in quadrant 4,
\n" ); document.write( "and the angle is \"360%5Eo\" minus
\n" ); document.write( "the quadrant 1 angle.
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\n" ); document.write( "FORMULAS:
\n" ); document.write( "Any complex number,
\n" ); document.write( " \"a+%2B+i+b\" ,
\n" ); document.write( "with \"a\" , and \"b\" being real numbers,
\n" ); document.write( "can always be written as
\n" ); document.write( " \"r+%28cos%28theta%29+%2B+i+sin%28theta%29%29\"
\n" ); document.write( "The formulas for \"r\" and \"theta\" are:
\n" ); document.write( "\"r+=+sqrt+%28a%5E2+%2B+b%5E2%29\"
\n" ); document.write( " \"cos%28theta%29+=+a%2Fr\"
\n" ); document.write( "and
\n" ); document.write( " \"sin%28theta%29++=+b%2Fr\" .
\n" ); document.write( "The last 2 formulas allow you to find theta
\n" ); document.write( "(in degrees or radians) with any scientific calculator.
\n" ); document.write( "If the calculator gives you a negative angle, add \"360%5Eo\" (or \"2pi\" when working in radians).
\n" ); document.write( "(For faster calculations, you can use
\n" ); document.write( "\"tan%28theta%29+=+b%2Fa\" and the signs, positive or negative, of \"a\" and \"b\" ). \n" ); document.write( "
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