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document.write( "Find the values of Arcsin 2x in the equation :
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document.write( "Arcsin 2x - Arcsin x = pi/3
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document.write( "Let \"a\" be the angle - 
<= a <= , sin(a) = x, and \r\n" );
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document.write( "Let \"b\" be the angle - <= b <= , sin(b) = 2x.\r\n" );
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document.write( "They want you find the value of x such that\r\n" );
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document.write( " b - a = 
. (1)\r\n" );
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document.write( "Since sin(a) = x and sin(b) = 2x, we have cos(a) = 
, cos(b) = 
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document.write( "From equation (1), taking cosine from both sides, we have this equation\r\n" );
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document.write( " cos(b-a) = 
, or\r\n" );
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document.write( " cos(b)*cos(a) + sin(b)*sin(a) = 
, or, substituting\r\n" );
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document.write( " 
+ (2x)*x = , or\r\n" );
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document.write( " 
= - 2x^2.\r\n" );
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document.write( "Now square both sides. You will get then\r\n" );
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document.write( " (1-4x^2)*(1-x^2) = 
- 2x^2 + 4x^4.\r\n" );
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document.write( "Simplify it step by step\r\n" );
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document.write( " 1 - 4x^2 - x^2 + 4x^4 = - 2x^2 + 4x^4\r\n" );
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document.write( " 1 - 5x^2 = - 2x^2\r\n" );
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document.write( " 4 - 20x^2 = 1 - 8x^2\r\n" );
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document.write( " 4 - 1 = 20x^2 - 8x^2\r\n" );
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document.write( " 3 = 12x^2\r\n" );
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document.write( " 1 = 4x^2\r\n" );
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document.write( " x^2 = \r\n" );
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document.write( " x = 
= +/- .\r\n" );
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document.write( "Thus the equation is just solved, and we have two potential solutions x= +/- .\r\n" );
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document.write( " Consider these two cases separately and check the results in both cases.\r\n" );
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document.write( "Case a). x = ; arcsin(x) = 
; 2x = 1; arcsin(2x) = arcsin(1) = .\r\n" );
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document.write( " Since - = 
= 
= , the solution is correct.\r\n" );
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document.write( "Case b). x = - ; arcsin(x) = - ; 2x = -1; arcsin(2x) = arcsin(-1) = - .\r\n" );
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document.write( " Since 
- 
= 
= - = - , this solution DOES NOT work. It is EXTRANEOUS.\r\n" );
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document.write( " | So, the problem has a unique solution |\r\n" );
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document.write( " | x = , and arcsin(x) = , arcsin(2x) = . | \r\n" );
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document.write( "Solved.\r
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document.write( "Post-solution notes:\r
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document.write( "1) The solution under the link\r\n" );
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document.write( " https://socratic.org/questions/how-do-you-solve-arcsin-x-arcsin-2x-pi-3\r\n" );
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document.write( " mentioned by tutor @Math_tutor2020, is INCORRECT.\r\n" );
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document.write( "2) The solution by @Math_tutor2020 also contains a technical error.\r\n" );
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document.write( " It is WHY I came to bring the correct and accurate solution.\r\n" );
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
