document.write( "Question 1195713: sin^-1(2x√1-x^2)+sin^-1(3x -4x^3)=-pi/3 the value of x \n" ); document.write( "

Algebra.Com's Answer #828230 by ikleyn(52909) $\"\"$ $\"About$

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\n" ); document.write( "sin^-1(2x√1-x^2)+sin^-1(3x -4x^3)=-pi/3 the value of x
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document.write( "Notice that due to the formula  sin(2a) = 2*sin(a)*cos(a),\r\n" );
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document.write( "you have  arcsin((2x*sqrt(1-x^2)) = 2a,  where  sin(a) = x.\r\n" );
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document.write( "Similarly, due to the formula  sin(3a) = 3*sin(a) - 4*sin^3(a),\r\n" );
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document.write( "you have  arcsin(3x-4x^3) = 3a,  where  sin(a) = x.\r\n" );
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document.write( "Therefore, the given equation\r\n" );
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document.write( "    sin^(-1)(2x*sqrt(1-x^2)) + sin^(-1)(3x -4x^3) = -pi/3\r\n" );
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document.write( "is equivalent to \r\n" );
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document.write( "    2a + 3a = -pi/3,  where  sin(a) = x,\r\n" );
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document.write( "or\r\n" );
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document.write( "      5a    = -pi/3,   sin(a) = x,\r\n" );
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document.write( "       a    =  =  = -12°.\r\n" );
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document.write( "It implies  x = .   It is the \"exact\" formula and the \"exact\" answer.\r\n" );
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document.write( "To get the numerical value, use your calculator or table of sinus: x = -0.20791169081... = -0.2079 (rounded).\r\n" );
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document.write( "CHECK.  2x*sqrt(1-x^2) =  = -0.406715;  arcsin(-0.406715) = -0.418855117.\r\n" );
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document.write( "        3x -4x^3 = 3*(-0.2079) - 4*(-0.2079)^3 = -0.587756244; arcsin(-0.587756244) = -0.628282675.\r\n" );
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document.write( "        -0.418855117 - -0.628282675 = -1.047137791.\r\n" );
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document.write( "        From the other side,   =  = -1.04719755.\r\n" );
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document.write( "        Both values,  -1.047137791  and  -1.04719755,  are close enough, confirming validity of the answer.\r\n" );
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