document.write( "Question 1191003: Prove that : cos(72°) =(1/2)sqrt(4sin²(36°)-1) \n" ); document.write( "
Algebra.Com's Answer #822993 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "She used special exact values for certain trig ratios for certain angles.\r\n" );
document.write( "I didn't use them. I begin by drawing an isosceles triangle with a vertex\r\n" );
document.write( "angle of 36o and base angles 72o each.\r\n" );
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document.write( "Now draw the internal bisector BD of the 72o angle B on the right \r\n" );
document.write( "into two 36o angles. Triangle ABD is isosceles since its base\r\n" );
document.write( "angles are both 36o. Then the linear pair of angles at D are 72o and 108o. \r\n" );
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document.write( "ΔDAB ∽ ΔABC because their internal angles are congruent.\r\n" );
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document.write( "Let AD = 1 and AB = x. Then by isosceles triangles\r\n" );
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document.write( "AB = DB = CD = x\r\n" );
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document.write( "By similar triangles ΔDAB and ΔABC,\r\n" );
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document.write( "\"AB%2FAD=AC%2FAB\", or\r\n" );
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document.write( "\"x%2F1=%281%2Bx%29%2Fx\"\r\n" );
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document.write( "\"x%5E2=1%2Bx\"\r\n" );
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document.write( "\"x%5E2-x-10\"\r\n" );
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document.write( "Solving for x by the quadratic formula:\r\n" );
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document.write( "\"x+=+%281+%2B+sqrt%285%29%29%2F2+\"\r\n" );
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document.write( "We must use the plus sign since the minus sign\r\n" );
document.write( "gives a negative value for x.\r\n" );
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document.write( "By the law of cosines on ΔDAB,\r\n" );
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document.write( "\"BD%5E2=AB%5E2%2BAD%5E2-2%2AAB%2AAD%2Acos%28A%29\"\r\n" );
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document.write( "\"x%5E2=x%5E2%2B1%5E2-2%2Ax%2A1%2Acos%2872%5Eo%29\" \r\n" );
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document.write( "\"2x%2Acos%2872%5Eo%29=1\"\r\n" );
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document.write( "\"cos%2872%5Eo%29=1%2F%282x%29=1%2F%282%28%281%2Bsqrt%285%29%29%2F2%29%29\"\r\n" );
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document.write( "Simplifying and rationalizing the denominator,\r\n" );
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document.write( "\"cos%2872%5Eo%29=%28sqrt%285%29-1%29%2F4%29%29\"\r\n" );
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document.write( "I will use that for the left side of the identity.\r\n" );
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document.write( "By the law of sines on ΔDAB,\r\n" );
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document.write( "\"AD%2Fsin%28%22%22%3CABD%29=BD%2Fsin%28A%29\"\r\n" );
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document.write( "\"1%2Fsin%2836%5Eo%29=x%2Fsin%2872%5Eo%29\"\r\n" );
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document.write( "\"sin%2872%5Eo%29=x%2Asin%2836%5Eo%29\"\r\n" );
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document.write( "\"sqrt%281-cos%5E2%2872%5Eo%29%29=%28%281+%2B+sqrt%285%29%29%2F2%29sin%2836%5Eo%29\"\r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "\"2sqrt%281-cos%5E2%2872%5Eo%29%29=%281+%2B+sqrt%285%29%29sin%2836%5Eo%29\"\r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "\"4%281-cos%5E2%2872%5Eo%29%29=%281+%2B+sqrt%285%29%29%5E2%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"4-4cos%5E2%2872%5Eo%29=%281+%2B+2sqrt%285%29%2B5%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"4-4%28%28sqrt%285%29-1%29%2F4%29%5E2=%286+%2B+2sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"4-4%285-2sqrt%285%29%2B1%29%2F16=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"4-%286-2sqrt%285%29%29%2F4=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"4-2%283-sqrt%285%29%29%2F4=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"4-%283-sqrt%285%29%29%2F2=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "Multiply through by 2\r\n" );
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document.write( "\"8-%283-sqrt%285%29%29=4%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"8-3%2Bsqrt%285%29=4%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"5%2Bsqrt%285%29=4%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "\"%285%2Bsqrt%285%29%29%2F%284%283+%2B+sqrt%285%29%29%29=sin%5E2%2836%5Eo%29\"\r\n" );
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document.write( "Rationalize the denominator:\r\n" );
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document.write( "\"sin%5E2%2836%5Eo%29=%285-sqrt%285%29%29%2F8\"\r\n" );
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document.write( "I will use that on the right side of the original identity\r\n" );
document.write( "which is to be proved:\r\n" );
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document.write( "\"cos%2872%5Eo%29+=%281%2F2%29sqrt%284sin%5E2%2836%5Eo%29-1%29\"\r\n" );
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document.write( "\"%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%284%28%285-sqrt%285%29%29%2F8%29-1%29\"\r\n" );
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document.write( "\"%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%28%285-sqrt%285%29%29%2F2%29-1%29\"\r\n" );
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document.write( "\"%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%28%285-sqrt%285%29%29%2F2%29-2%2F2%29\"\r\n" );
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document.write( "\"%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%285-sqrt%285%29-2%29%2F2%29\"\r\n" );
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document.write( "\"%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "Since the left side is positive, it is the positive square \r\n" );
document.write( "root of its square:\r\n" );
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document.write( "\"sqrt%28%28+%28sqrt%285%29-1%29%2F4%29%5E2+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "\"sqrt%28%28+%286-2sqrt%285%29%29%2F16%29+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "\"sqrt%28%28+2%283-sqrt%285%29%29%2F16%29+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "\"sqrt%28+%283-sqrt%285%29%29%2F8+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "\"sqrt%28+%283-sqrt%285%29%29%2F%284%2A2%29+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "\"%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29\"\r\n" );
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document.write( "Edwin
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