document.write( "Question 1009489: Solve the given equation for x
\n" ); document.write( "2 arcsin(x) + arccos(x) = π \n" ); document.write( "

Algebra.Com's Answer #625094 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "arcsin(x) means \"The angle whose sine is x between  and  \r\n" );
document.write( "arccos(x) means \"The angle whose cosine is x\" between  and \r\n" );
document.write( "\"CoSine\" means \"Complement's Sine \r\n" );
document.write( "We must consider two cases.  \r\n" );
document.write( "Case 1: when x is positive\r\n" );
document.write( "Then the arcsin(x) and arcos(x) are both in QI\r\n" );
document.write( "So x will be  \r\n" );
document.write( "90°-the angle whose sine is x.\r\n" );
document.write( "But since we are using radians instead of degrees, it is:\r\n" );
document.write( " - the angle whose sine is x. \r\n" );
document.write( "So \r\n" );
document.write( "and therefore\r\n" );
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document.write( "becomes\r\n" );
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document.write( "Multiply through by LCD of 2 to clear the fraction:\r\n" );
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document.write( "Therefore the Case 1 solution is x=1 since \r\n" );
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document.write( "Case 2: when x is negative\r\n" );
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document.write( "Then the arcsin(x) is a negative angle in QIV \r\n" );
document.write( "And so 2arcsin(x) is an even more negative angle than arcsin(x)\r\n" );
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document.write( "arccos(x) is a positive angle in QII less than \r\n" );
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document.write( "So 2arcsin(x)+cos(x) is the sum of a positive angle less than\r\n" );
document.write( "pi, and a negative angle.  The sum of a positive angle less \r\n" );
document.write( "than  added to a negative angle can never equal to , \r\n" );
document.write( "so there is no solution to case 2.\r\n" );
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document.write( "Thus the only solution is x=1.\r\n" );
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document.write( "Edwin

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