document.write( "Question 868206: How do I find the exact value of $\"y+=+cos%284pi%2F5%29\"$ ? \n" ); document.write( "
Algebra.Com's Answer #523699 by Edwin McCravy(20059) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Let A = \r\n" );
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document.write( "Then  must be one of the solutions to:\r\n" );
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document.write( "Use identity \r\n" );
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document.write( "Use identity  on \r\n" );
document.write( "Use identity  on \r\n" );
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document.write( "Use identity  on \r\n" );
document.write( "Use identity  on \r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "Factor sin²(A) out of the last two terms on the left:\r\n" );
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document.write( "Use the identity  on sin²(A)\r\n" );
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document.write( "FOIL out the last term on the left:\r\n" );
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document.write( "Let cos(A)=x\r\n" );
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document.write( "Try a root of 1\r\n" );
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document.write( "1 | 16   0  -20   0   5  -1\r\n" );
document.write( "  |     16   16  -4  -4   1\r\n" );
document.write( "    16  16   -4  -4   1   0\r\n" );
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document.write( "So it factors as\r\n" );
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document.write( "We try factoring  as\r\n" );
document.write( "the product of two quadratics.  In fact we might\r\n" );
document.write( "expect that it is the square of a quadratic because\r\n" );
document.write( "the leading term  is a square and\r\n" );
document.write( "also the constant term 1 is a square.\r\n" );
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document.write( "So we try to see if there is a quadratic (4x²+kx-1)\r\n" );
document.write( "which when squared gives  \r\n" );
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document.write( "Equating coefficients of x³:\r\n" );
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document.write( "8k = 16\r\n" );
document.write( "  k = 2\r\n" );
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document.write( "Equating coefficients of x²:\r\n" );
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document.write( "k²-8 = -4\r\n" );
document.write( "   k² = 4\r\n" );
document.write( "    k = ±2\r\n" );
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document.write( "So far, k=2 works\r\n" );
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document.write( "Equating  coefficients of x\r\n" );
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document.write( "-2k=-4\r\n" );
document.write( "   k=2\r\n" );
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document.write( "So indeed \r\n" );
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document.write( "factors as\r\n" );
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document.write( "So we have solutions\r\n" );
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document.write( "x-1 = 0    and   \r\n" );
document.write( "  x = 1           \r\n" );
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document.write( "So the solutions are\r\n" );
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document.write( ",  , .\r\n" );
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document.write( "Since  is in Q2, its cosine must be negative.  The\r\n" );
document.write( "only solution which is negative is\r\n" );
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document.write( "thus\r\n" );
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document.write( "Edwin
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