document.write( "Question 885756: Determine analytically the value of 't' between 0 and π when \"5sin%28t%29+=+8sin%28t%2Bpi%2F8%29\"\r
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Algebra.Com's Answer #535515 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"5sin%28t%29+=+8sin%28t%2Bpi%2F8%29\"\r\n" );
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document.write( "\"5sin%28t%29+=+8%28sin%28t%29cos%28pi%2F8%29%2Bcos%28t%29sin%28pi%2F8%29%29\"\r\n" );
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document.write( "\"5sin%28t%29+=+8sin%28t%29cos%28pi%2F8%29%2B8cos%28t%29sin%28pi%2F8%29%29\"\r\n" );
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document.write( "Use \"cos%28theta%2F2%29=+%22%22+%2B-+sqrt%281%2Bcos%28theta%29%29%2F2%29\" and \"sin%28theta%2F2%29=+%22%22+%2B-+sqrt%281-cos%28theta%29%29%2F2%29\" with \"theta=pi%2F4\"\r\n" );
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document.write( "We use only the positives since \"cos%28pi%2F8%29\" is positive.\r\n" );
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document.write( "Let \"A=sqrt%282%2Bsqrt%282%29%29\" and \"B=sqrt%282-sqrt%282%29%29\"\r\n" );
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document.write( "\"5sin%28t%29+=+4Asin%28t%29%2B4Bcos%28t%29\"\r\n" );
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document.write( "\"5sin%28t%29+-+4Asin%28t%29=4Bcos%28t%29\" \r\n" );
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document.write( "\"%285-+4A%29sin%28t%29=4Bcos%28t%29\"\r\n" );
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document.write( "Divide both sides by cos(t)\r\n" );
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document.write( "\"%285-+4A%29%2Aexpr%28sin%28t%29%2Fcos%28t%29%29=4B%2Aexpr%28cos%28t%29%2Fcos%28t%29%29\"\r\n" );
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document.write( "\"%285-+4A%29tan%28t%29=4B\"\r\n" );
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document.write( "\"tan%28t%29=4B%2F%285-4A%29\"\r\n" );
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document.write( "Since A is a radical we multiply top and bottom by the conjugate\r\n" );
document.write( "of the denominator, hoping to eventually rationalize the denominator.\r\n" );
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document.write( "We calculate AB and AČ\r\n" );
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document.write( "\"AB=+sqrt%282%2Bsqrt%282%29%29sqrt%282-sqrt%282%29%29=sqrt%284-2%29=sqrt%282%29\"\r\n" );
document.write( "\"A%5E2=+%28sqrt%282%2Bsqrt%282%29%29%29%5E2=2%2Bsqrt%282%29\"\r\n" );
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document.write( "Rationalizing further:\r\n" );
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document.write( "\"tan%28t%29=%284%285B%287-16sqrt%282%29%29%2B28sqrt%282%29-128%29%29%2F463++\"\r\n" );
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document.write( "\"tan%28t%29+=+-1.280393572\", approximately\r\n" );
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document.write( "Finding the inverse tangent of the absolute value of that gives .9077424765\r\n" );
document.write( "radians.  But since the tangent is negative, t must be in the 2nd and\r\n" );
document.write( "4th quadrants.  But yu are only asking for the 2nd quadrant answer, since\r\n" );
document.write( "the interval given is between 0 and \"pi\".\r\n" );
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document.write( "To get the second quadrant answer subtract from \"pi\".\r\n" );
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document.write( "\"t=pi-.9077424765=2.233850177\"\r\n" );
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document.write( "Edwin
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