document.write( "Question 1035075: If ( cos x+ isin x )(cos y+ isin y) = cos (y+x) + isin (y+x)
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Algebra.Com's Answer #649692 by ikleyn(52908)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "If ( cos x+ isin x )(cos y+ isin y) = cos (y+x) + isin (y+x)
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\n" ); document.write( "\n" ); document.write( "This formulation is incorrect and only can confuse the student.\r
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\n" ); document.write( "\n" ); document.write( "The correct formulation is \r
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\n" ); document.write( "\n" ); document.write( "(cos(x) + i*sin(x))*(cos(y)+ i*sin(y)) = cos(y+x) + i*sin(y+x), \r
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\n" ); document.write( "\n" ); document.write( "derive the formulas for \r
\n" ); document.write( "\n" ); document.write( "Sin (x+y)
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document.write( "OK. Let me start.\r\n" );
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document.write( "The formula \r\n" );
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document.write( "   (cos(x) + i*sin(x))*(cos(y)+ i*sin(y)) = cos(y+x) + i*sin(y+x)    (1)\r\n" );
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document.write( "is valid for any two complex numbers \r\n" );
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document.write( "   u = (cos(x) + i*sin(x)) and v = (cos(y) + i*sin(y))               (2)\r\n" );
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document.write( "that have the modulus 1 (are the \"unit\" vectors of the length 1 in the unit circle in the complex domain).\r\n" );
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document.write( "The \"x\" and \"y\" are \"arguments\" of the complex numbers \"u\" and \"v\".\r\n" );
document.write( "Geometrically, it means that \"x\" and \"y\" are the angles on the unit circle.\r\n" );
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document.write( "This formula and the complex numbers came from the section of the school math that is called \"Complex numbers\". \r\n" );
document.write( "So, I assume that you have the minimal introductory knowledge to understand what I am writing.\r\n" );
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document.write( "Actually, there is a block of my lessons on complex numbers in this site, that might be useful for you.\r\n" );
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document.write( "The list of these lessons is at the end of my post, and you can look into them later. Right now I will continue.\r\n" );
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document.write( "Let us perform this multiplication of the two complex numbers in the left side of (1). \r\n" );
document.write( "Using the rules for multiplication of complex numbers, you have\r\n" );
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document.write( "  (cos(x) + i*sin(x))*(cos(y)+ i*sin(y)) = \r\n" );
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document.write( "= (cos(x)*cos(y) - sin(x)*sin(y))  +  i*(sin(x)*cos(y) + cos(x)*sin(y).   (3)\r\n" );
document.write( "   ----------------------------         -----------------------------\r\n" );
document.write( "    This is the real part                And this is the imaginary\r\n" );
document.write( "       of the product                       part of the product\r\n" );
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document.write( "Now compare it with the right side of the formula (1).\r\n" );
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document.write( "The complex number in the left side of (1), which is (3) is equal to the complex number in the right side of (1).\r\n" );
document.write( "It means that the real parts of (3) and (1) are equal, as well as the imaginary parts of (3) and (1) are equal too.\r\n" );
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document.write( "In other words,\r\n" );
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document.write( "   cos(x)*cos(y) - sin(x)*sin(y) = cos(x+y),    (4)\r\n" );
document.write( "   sin(x)*cos(y) + cos(x)*sin(y) = sin(x+y).    (5)\r\n" );
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document.write( "So, based on the equality (1) for complex numbers, you got these formulas for angles \"x\" and\"y\"\r\n" );
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document.write( "   cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)   and\r\n" );
document.write( "   sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y).\r\n" );
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document.write( "If you studied Trigonometry before, you can recognize that the last two formulas are the \"addition formulas for cosine and sine\".\r\n" );
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document.write( "That is all my story for now.\r\n" );
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document.write( "At the end, as I promised, get the list of my lessons on complex numbers in this site:\r\n" );
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document.write( "    - Complex numbers and arithmetic operations on them\r\n" );
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document.write( "    - Complex plane\r\n" );
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document.write( "    - Addition and subtraction of complex numbers in complex plane\r\n" );
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document.write( "    - Multiplication and division of complex numbers in complex plane\r\n" );
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document.write( "    - Raising a complex number to an integer power\r\n" );
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document.write( "    - How to take a root of a complex number\r\n" );
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document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r\n" );
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document.write( "    - How to take a square root of a complex number\r\n" );
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document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r\n" );
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