document.write( "Question 515029: Write the complex number z=-4-5i in trigonometric form (sometimes called polar form), with the angle expressed in radians from 0 to 2pi. Do not round any intermediate computations, and round the values in your answer to 2decimal places\r
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Algebra.Com's Answer #343695 by Edwin McCravy(20063) $\"\"$ $\"About$

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Write the complex number z=-4-5i in trigonometric form (sometimes called polar form), with the angle expressed in radians from 0 to 2pi. Do not round any intermediate computations, and round the values in your answer to 2decimal places
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document.write( "We want it in the form -4-5i = R(cos + i*sin)\r\n" );
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document.write( "First let's draw the complex number -4-5i. It is the vector drawn from \r\n" );
document.write( "(0,0) to the point (-4,-5).  Here it is (in green):\r\n" );
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document.write( "Now let's draw a vertical line from the tip of the\r\n" );
document.write( "vector z to the x-axis  (in blue):\r\n" );
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document.write( "So we have x = -4, y = -5\r\n" );
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document.write( "We calculate R from R = \r\n" );
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document.write( "Now that we have R, we only need to find , which is the\r\n" );
document.write( "angle marked  below and indicated by the red arc, rotating\r\n" );
document.write( "from the right side of the x-axis counter-clockwise around to the \r\n" );
document.write( "green vector. \r\n" );
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document.write( "However the calculator will not give us that value for .\r\n" );
document.write( "So we get the reference angle, which is indicated by the blue arc\r\n" );
document.write( "below:\r\n" );
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document.write( "Now we use our calculator to find the inverse tangent of , and\r\n" );
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document.write( "find the angle indicated by the blue are (called the \"reference angle\"\r\n" );
document.write( " to be 0.8960553846 radians.\r\n" );
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document.write( "However we must add  to that number to indicate the complete\r\n" );
document.write( "angle .  So we have  =  + 0.8960553846\r\n" );
document.write( "= 4.037648038, which rounds to 4.04 radians.  Therefore the \r\n" );
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document.write( "-4-5i = R(cos + i*sin) = (cos4.04 + i*sin4.04).  You can also evaluate  as 6.403124237\r\n" );
document.write( "and make the answer \r\n" );
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document.write( "6.40(cos4.04 + i*sin4.04)\r\n" );
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document.write( "Edwin

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