document.write( "Question 577079: Preform the indicated operation using trigonometric form. Leave answer in trigonometric form
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Algebra.Com's Answer #370109 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
(-3-3i)(2+2i)
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document.write( "Convert each to trig form:\r\n" );
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document.write( "First we convert -3-3i to trig form:\r\n" );
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document.write( "1.  The complex number x+yi is represented by the radius vector (line segment)\r\n" );
document.write( "    from the origin (0,0) to the point (x,y).  So draw the radius vector, and\r\n" );
document.write( "    the perpendicular from the point (x,y) to the x-axis.\r\n" );
document.write( "2.  Calculate the length of that vector r, using the Pythagorean theorem:\r\n" );
document.write( "    r²=x²+y².  That value is called the modulus of the complex number.\r\n" );
document.write( "3.  Calculate the angle q from the right side of the axis around to the\r\n" );
document.write( "    radius vector.  To do this you may use any of the trig ratios involving\r\n" );
document.write( "    x, y and r.  This angle is called the argument. \r\n" );
document.write( "4.  Write the trig form as r(cosq + i·sinq)\r\n" );
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document.write( "First we convert -3-3i to trig form:\r\n" );
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document.write( "1.  We draw the radius vector connecting the origin to the point (-3,-3) and\r\n" );
document.write( "    the perpendicular from that point to the to the x-axis.  We label the\r\n" );
document.write( "    perpendicular to the x-axis as y=-3 and the segment from the origin to\r\n" );
document.write( "    the perpendicular.  We label the length of the radius vector r, and\r\n" );
document.write( "    indicate the argument q with a counter-clockwise\r\n" );
document.write( "    arc from the right side of the x-axis around to the radius vector:       \r\n" );
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document.write( "2.  We calculate the length of that vector r, using the Pythagorean theorem:\r\n" );
document.write( "    r²=x²+y².  \r\n" );
document.write( "    r²=(-3)²+(-3)²\r\n" );
document.write( "    r²=9+9\r\n" );
document.write( "    r²=18\r\n" );
document.write( "     r=\"sqrt%2818%29\"\r\n" );
document.write( "     r=\"sqrt%289%2A2%29\"\r\n" );
document.write( "     r = 3\"sqrt%282%29\" \r\n" );
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document.write( "3.  We calculate the angle q, by realizing that\r\n" );
document.write( "    the right-triangle is a 45°-45°-90° with a reference angle of 45°, and\r\n" );
document.write( "    the actual angle q = 180°+45° = 225°\r\n" );
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document.write( "4.  We write the trig form as r(cosq + i·sinq), or \r\n" );
document.write( "    \r\n" );
document.write( "    \"3sqrt%282%29\"(cos225° + i·sin225°).\r\n" );
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document.write( "Next we convert 2+2i to trig form:\r\n" );
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document.write( "1.  We draw the radius vector connecting the origin to the point (2,2) and\r\n" );
document.write( "    the perpendicular from that point to the to the x-axis.  We label the\r\n" );
document.write( "    perpendicular to the x-axis as y=2 and the segment from the origin to\r\n" );
document.write( "    the perpendicular x=2.  We label the length of the radius vector r, and\r\n" );
document.write( "    indicate the argument q with a counter-clockwise\r\n" );
document.write( "    arc from the right side of the x-axis around to the radius vector:       \r\n" );
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document.write( "2.  We calculate the length of that vector r, using the Pythagorean theorem:\r\n" );
document.write( "    r²=x²+y².  \r\n" );
document.write( "    r²=(2)²+(2)²\r\n" );
document.write( "    r²=4+4\r\n" );
document.write( "    r²=8\r\n" );
document.write( "     r=\"sqrt%288%29\"\r\n" );
document.write( "     r=\"sqrt%284%2A2%29\"\r\n" );
document.write( "     r = 2\"sqrt%282%29\" \r\n" );
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document.write( "3.  We calculate the angle q, by realizing that\r\n" );
document.write( "    the right-triangle is also a 45°-45°-90° which is an angle of 45°\r\n" );
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document.write( "4.  We write the trig form as r(cosq + i·sinq), or \r\n" );
document.write( "    \r\n" );
document.write( "    \"2sqrt%282%29\"(cos45° + i·sin45°).\r\n" );
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document.write( "Now we use the formula for multiplying complex numbers in trig form:\r\n" );
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document.write( "  r1(cosq1 + i·sinq1)·r2(cosq2 + i·sinq2) =   r1r2[cos(q1+q2) + i·sin(q1+q2)]. \r\n" );
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document.write( "So we have:\r\n" );
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document.write( "(-3-3i)(2+2i) = \"3sqrt%282%29\"(cos225° + i·sin225°)·\"2sqrt%282%29\"(cos45° + i·sin45°) =\r\n" );
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document.write( "\"2sqrt%282%29%2A3sqrt%282%29\"[cos(225°+45°) + i·sin(225°+45°)] = 6·2[cos270° + i·sin270°] =\r\n" );
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document.write( "12(cos270° + i·sin270°).\r\n" );
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document.write( "Edwin
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