document.write( "Question 799298: Find the polar representation of [sin(a)+cos(a)] + i[sin(a)-cos(a)]
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Algebra.Com's Answer #482664 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "[sin(a)+cos(a)] + i[sin(a)-cos(a)]\r\n" );
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document.write( "We use the facts that\r\n" );
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document.write( "1. cos(X)cos(Y)+cos(X)sin(Y) = cos(X-Y)\r\n" );
document.write( "2. sin(X)cos(Y)-cos(X)sin(Y) = sin(X-Y)\r\n" );
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document.write( "and\r\n" );
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document.write( "3. sin(\"pi%2F4\") = cos(\"pi%2F4\") = \"1%2Fsqrt%282%29\"\r\n" );
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document.write( "Let's take the real part first:\r\n" );
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document.write( "sin(a)+cos(a), write it over 1, \"%28sin%28a%29%2Bcos%28a%29%29%2F1\"   \r\n" );
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document.write( "Multiply it by \"cos%28pi%2F4%29%2Fcos%28pi%2F4%29\" which just equals 1\r\n" );
document.write( "and therefore will not change the value:\r\n" );
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document.write( "\"cos%28pi%2F4%29%2Fcos%28pi%2F4%29\"\"%22%22%2A%22%22\"\"%28sin%28a%29%2Bcos%28a%29%29%2F1\"\r\n" );
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document.write( "\"%28+cos%28pi%2F4%29%28sin%28a%29%2Bcos%28a%29%29%29%2Fcos%28pi%2F4%29\" \r\n" );
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document.write( "Distribute on top \r\n" );
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document.write( "\"%28+cos%28pi%2F4%29sin%28a%29%2Bcos%28pi%2F4%29cos%28a%29+%29%2Fcos%28pi%2F4%29\"\r\n" );
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document.write( "By 3 above, replace the first cos(\"pi%2F4\") by sin(\"pi%2F4\")\r\n" );
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document.write( "\"%28+sin%28pi%2F4%29sin%28a%29%2Bcos%28pi%2F4%29cos%28a%29+%29%2Fcos%28pi%2F4%29\"\r\n" );
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document.write( "Rearrange to look like cos(X)cos(Y)+cos(X)sin(Y) = cos(X-Y)\r\n" );
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document.write( "\"%28+cos%28a%29cos%28pi%2F4%29%2Bsin%28a%29sin%28pi%2F4%29+%29%2Fcos%28pi%2F4%29\" \r\n" );
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document.write( "So the numerator becomes cos(a-\"pi%2F4\")\r\n" );
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document.write( "\"cos%28a-pi%2F4%29%2Fcos%28pi%2F4%29\" \r\n" );
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document.write( "Since the denominator \"cos%28pi%2F4%29\" = \"1%2Fsqrt%282%29\", we substitute \r\n" );
document.write( "and get:\r\n" );
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document.write( "\"cos%28a-pi%2F4%29%2F%281%2Fsqrt%282%29%29\" = \"cos%28a-pi%2F4%29\"\"%22%F7%22\"\"1%2Fsqrt%282%29\" = \"cos%28a-pi%2F4%29\"\"%22%22%2A%22%22\"\"cos%28a-pi%2F4%29\" \r\n" );
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document.write( "Now we take the imaginary part, the coefficient of i:\r\n" );
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document.write( "sin(a)-cos(a), write it over 1, \"%28sin%28a%29-cos%28a%29%29%2F1\"   \r\n" );
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document.write( "Multiply it by \"sin%28pi%2F4%29%2Fsin%28pi%2F4%29\" which just equals 1\r\n" );
document.write( "and therefore will not change the value:\r\n" );
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document.write( "\"sin%28pi%2F4%29%2Fsin%28pi%2F4%29\"\"%22%22%2A%22%22\"\"%28sin%28a%29-cos%28a%29%29%2F1\"\r\n" );
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document.write( "\"%28+sin%28pi%2F4%29%28sin%28a%29-cos%28a%29%29%29%2Fsin%28pi%2F4%29\" \r\n" );
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document.write( "Distribute on top \r\n" );
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document.write( "\"%28+sin%28pi%2F4%29sin%28a%29-sin%28pi%2F4%29cos%28a%29+%29%2Fsin%28pi%2F4%29\"\r\n" );
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document.write( "By 3 above, replace the first sin(\"pi%2F4\") by cos(\"pi%2F4\")\r\n" );
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document.write( "\"%28+cos%28pi%2F4%29sin%28a%29-sin%28pi%2F4%29cos%28a%29+%29%2Fsin%28pi%2F4%29\"\r\n" );
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document.write( "Rearrange to look like sin(X)cos(Y)-cos(X)sin(Y) = sin(X-Y)\r\n" );
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document.write( "\"%28+sin%28a%29cos%28pi%2F4%29-cos%28a%29sin%28pi%2F4%29+%29%2Fsin%28pi%2F4%29\" \r\n" );
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document.write( "So the numerator becomes sin(a-\"pi%2F4\")\r\n" );
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document.write( "\"sin%28a-pi%2F4%29%2Fsin%28pi%2F4%29\" \r\n" );
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document.write( "Since the denominator \"sin%28pi%2F4%29\" = \"1%2Fsqrt%282%29\", we substitute \r\n" );
document.write( "and get:\r\n" );
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document.write( "\"sin%28a-pi%2F4%29%2F%281%2Fsqrt%282%29%29\" = \"sin%28a-pi%2F4%29\"\"%22%F7%22\"\"1%2Fsqrt%282%29\" = \"sin%28a-pi%2F4%29\"\"%22%22%2A%22%22\"\"sin%28a-pi%2F4%29\" \r\n" );
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document.write( "So the original problem \r\n" );
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document.write( "[sin(a)+cos(a)] + i[sin(a)-cos(a)]\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "\"sqrt%282%29\"\"cos%28a-pi%2F4%29\" + i·\"sqrt%282%29\"\"sin%28a-pi%2F4%29\" \r\n" );
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document.write( "\"sqrt%282%29\"[\"cos%28a-pi%2F4%29\" + i·\"sin%28a-pi%2F4%29\"] \r\n" );
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document.write( "That's the polar representation\r\n" );
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document.write( "which is often written as \r\n" );
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document.write( "\"sqrt%282%29\"\"cis%28a-pi%2F4%29\",\r\n" );
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document.write( "and electrical engineers\r\n" );
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document.write( "write it as \"sqrt%282%29\"∠\"%28a-pi%2F4%29\".  \r\n" );
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document.write( " Edwin
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