document.write( "Question 476982: What do you call this formula z=|z| (cosѲ + sinѲ)?
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Algebra.Com's Answer #327034 by Edwin McCravy(20064)\"\" \"About 
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document.write( "You left out the i.  It's\r\n" );
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document.write( "z = |z|(cosѲ + i·sinѲ)\r\n" );
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document.write( "That's the trigonometric form of the complex number z = x + iy\r\n" );
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document.write( "\"z=x%2Biy=+abs%28z%29%28cos%28theta%29+%2B+i%2Asin%28theta%29%29\"\r\n" );
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document.write( "The complex imaginary number z = x + yi is represented \r\n" );
document.write( "by the vector (line segment) from the origin to the point\r\n" );
document.write( "(x,y).  So we draw and arbritrary point (x,y). (x,y) could \r\n" );
document.write( "be in any quadrant or even on an axis, but we'll draw it in \r\n" );
document.write( "the first quadrant. We'll label the length of the vector \r\n" );
document.write( "from the origin to the point (x,y) as |z|. \r\n" );
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document.write( "Now from the point (x,y) we drop a perpendicular to the\r\n" );
document.write( "x-axis (in green):\r\n" );
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document.write( "That makes a right triangle.  The base of that right triangle\r\n" );
document.write( "is the same as the x-coordinate of the point (x,y), so it is x.\r\n" );
document.write( "The green side of that right triangle\r\n" );
document.write( "is the same as the y-coordinate of the point (x,y), so it is y.\r\n" );
document.write( "The angle that the vector makes with the x-axis is labeled Ѳ.\r\n" );
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document.write( "So from the triangle,\r\n" );
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document.write( "\"cos%28theta%29=x%2Fabs%28z%29\" which gives \"x+=+abs%28z%29cos%28theta%29\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"sin%28theta%29=y%2Fabs%28z%29\" which gives \"y+=+abs%28z%29sin%28theta%29\"\r\n" );
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document.write( "Therefore\r\n" );
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document.write( "\"z=x%2Biy+=+abs%28z%29cos%28theta%29+%2B+i%2Aabs%28z%29sin%28theta%29\"\r\n" );
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document.write( "Then we can factor out the \"abs%28z%29\" and get\r\n" );
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document.write( "\"z=x%2Biy+=+abs%28z%29%28cos%28theta%29+%2B+i%2Asin%28theta%29%29\"\r\n" );
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document.write( "That's the trigonometric or \"trig\" form of the complex \r\n" );
document.write( "number x+iy.\r\n" );
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document.write( "It allows us to bring trig into the algebra of complex\r\n" );
document.write( "numbers.  It will make certain operations simpler. \r\n" );
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document.write( "Edwin
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