document.write( "Question 1005843: SOLVE THE EQUATION SIN3X=SINX \n" ); document.write( "

Algebra.Com's Answer #621993 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "SOLVE THE EQUATION SIN3X=SINX
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document.write( "Start with the formula of triple argument for sine:\r\n" );
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document.write( "sin(3x) = -4*sin^3(x) + 3*sin(x)\r\n" );
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document.write( "(see,  for example,  the lesson  Trigonometric functions of multiply argument  in this site).\r\n" );
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document.write( "It gives you an equation \r\n" );
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document.write( "-4*sin^3(x) + 3*sin(x) = sin(x).\r\n" );
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document.write( "Move the term sin(x) from the right side to the left with the opposite sign and then simplify.\r\n" );
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document.write( "-4*sin^3(x) + 3*sin(x) - sin(x) = 0,\r\n" );
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document.write( "-4*sin^3(x) + 2*sin(x) = 0.\r\n" );
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document.write( "Now factor the left side\r\n" );
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document.write( " = .   (1)\r\n" );
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document.write( "In this way the equation (1) decomposes in two equations. One is\r\n" );
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document.write( "sin(x) = 0 with the solutions x = 0, +/-, +/-, +/-, . . . , +/-, . . . (2)\r\n" );
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document.write( "The other equation is \r\n" );
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document.write( " = ,   or\r\n" );
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document.write( "sin(x) = +/- .          \r\n" );
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document.write( "It has the roots x = +/- .   (3)\r\n" );
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document.write( "So, the answer is the combination (the union) of solutions (2) and (3).\r\n" );
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document.write( "x = +/- {k*pi}}}, k = 0, 1, 2, 3, . . .  and\r\n" );
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document.write( "x = +/- , k = 0, 1, 2, 3, . . . \r\n" );
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document.write( "Obviously, the last set overlay the previous one, so you may restrict yourself by the last formula.\r\n" );
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