Algebra.Com's Answer #702542 by ikleyn(53763)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "If 3sinA + 5cosA = 5 then 5sinA - 3cosA equals
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document.write( "I will give ABSOLUTELY UNUSUAL and UNEXPECTED solution that you never saw/heard before.\r
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document.write( "1. You are given that \r\n" );
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document.write( " 3*sin(A) + 5*cos(A) = 5.\r\n" );
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document.write( " Divide both sides by  =  . You will get\r\n" );
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document.write( "  +  =  . (1)\r\n" );
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document.write( "2. Consider vector U = ( ,) and vector B = (sin(A),cos(A)).\r\n" );
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document.write( " The equality (1) means that the scalar product of these vectors is equal to .\r\n" );
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document.write( " In other words, \r\n" );
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document.write( " |U|*|B|*cos(a) = , (2)\r\n" );
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document.write( " where |U| is the modulus (= the magnitude, the length) of the vector U, |B| is the modulus of the vector B and \r\n" );
document.write( " \"a\" is the angle between these two vectors.\r\n" );
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document.write( " But (!!!) the length of the vector U is equal to 1, as well as the length of the vector B. \r\n" );
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document.write( " (Everybody who knows how to calculate the length of the vector, can easily check it).\r\n" );
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document.write( " Therefore, the equality (2) means that the cosine of the angle \"a\" is equal to :\r\n" );
document.write( " cos(a) = . (3)\r\n" );
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document.write( "3. Very good. Now let us consider the expression y = 5*sin(A) - 3*cos(A) which is under the question.\r\n" );
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document.write( " Again, divide both sides by . You will get\r\n" );
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document.write( "  =  . (4)\r\n" );
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document.write( " The right side of (4) is NOTHING ELSE AS the scalar product of the vectors V = (, ) and vector B introduced above.\r\n" );
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document.write( " Notice that the vector V has the length of 1 and is orthogonal to vector U.\r\n" );
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document.write( " So, V is the unit vector orthogonal to U.\r\n" );
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document.write( " Therefore, (4) simply is\r\n" );
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document.write( " = cos(b), (5)\r\n" );
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document.write( " where \"b\" is the angle between the vectors V and B.\r\n" );
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document.write( "4. BUT (!!!), since the vectors U and V are orthogonal, the angles \"a\" and \"b\" are complementary:\r\n" );
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document.write( " b =  - a.\r\n" );
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document.write( " Then \r\n" );
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document.write( " cos(b) = sin(a) =  =  =  =  =  = +/-. (6)\r\n" );
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document.write( " Thus = cos(b) = +/- .\r\n" );
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document.write( " It implies that y = +/- 3.\r\n" );
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document.write( "Answer. If 3*sin(A) + 5*cos(A) = 5 then 5*sin(A) - 3*cos(A) = +/- 3.\r
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document.write( "Solved.\r
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document.write( "Plots y1 = 3*sin(x) + 5*cos(x) (red), y2 = 5 (green), y3 = 5*sin(x) - 3*cos(x) (blue) and y4 = 3 (purple).\r
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document.write( "The plot shows very clearly that there are two points in the segment [ , ] where 3*sin(x) + 5*cos(x) = 5.\r
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document.write( "One of these points is x = A = 0. But there is the other point, too.\r
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document.write( "The plot shows very clearly, also, that at the points where 3*sin(x) + 5*cos(x) = 5 we have 5*sin(x) - 3*cos(x) = +/- 3.\r
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document.write( "It serves as the CHECK.\r
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