Trignometry

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how to find the value of sin 18 degree?
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Trigonometry

how to find the value of sin 18 degree?



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edison's Avatar

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19 Jan 2010 11:14:56 IST
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The Derivation of Sin 18°


 

  sin 72° = 2 sin 36° cos 36°                                 by the double angle relationship.
sin 72° = 4 sin 18° cos 18° (1 - 2sin2 18°)         by the double angle relationship, again.
cos 18° = 4 sin 18° cos 18° (1 - 2sin2 18°)         by the cofunction properties: sin 72° = cos 18°.
1 = 4 sin 18° (1 - 2sin2 18°)                       Let x = sin 18°, this is known as 
1 = 4x(1-2x2)                                             substitution, a useful technique in calculus.
8x3-4x+1 = 0                                                         A product is zero only when one of its factors is zero.
8x3-4x+1 = (2x-1)(4x2+2x-1)=0                           (2x-1)=0 implies x= ½=sin 30° > sin 18° ;
Since we know sin is increasing on [0°,90°].
x = (-2 ± [square root of] (4 + 4•4•1))/8                       So we must solve the other factor,
= (-2 ± [square root of] 20)/8                                     using the quadratic formula.
= (-2 ± [square root of] 4[square root of] 5)/8
= (-1 ± [square root of] 5)/4                                       But the sin 18° > 0, so it cannot be negative.
sin 18°   = ([square root of] (5)-1)/4                                         Hence the middle root is the one we want.

Dksha sharma's Avatar

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31 Jan 2010 14:38:45 IST
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let x=18" then,

5x=90"

2x+3x=90"

2x=90"-3x

sin2x=sin(90"-3x)

sin2x=cos3x                       {since sin (90" - x) =cos x}

2sinxcosx=4cos3x - cos x         {sin 2x= 2sinxcos x...and....cos 3x=4cos3 x-3cosx}

cos x(2sin x - 4 cos2x+3)=0

2 sinx-4 cos 2x +3=0

2 sin x-4(1- sin2x)+3=0

4 sin2x +2 sinx-1=0

hence

sin x=(5)1/2-1/4

sin 18=(5)1/2-1/4

 


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29 May 2010 14:56:05 IST
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The ratio between diagonal and side of any regular pentagon is  , the golden ratio = (1+ Square root of 5) / 2.

In a regular pentagon ABCDE, draw a perpendicular from A on to CD to meet it at F when angle FAD = angle FAC = 18 deg.

Sin 18 = half side / diagonal = 1/2 of [2/(1+ Square root of 5)] = [(Square root of 5)-1]/4

posted by : K SUBBA REDDY 63 years mobile: 9948711151


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16 Nov 2011 23:13:51 IST
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how to find the value of sin 18 degree?

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16 Nov 2011 23:19:41 IST
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no post plz on this blog


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26 Feb 2012 20:59:38 IST
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how to find the value of sin 18 degree?

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10 Apr 2012 12:41:38 IST
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how do we factorise 8x3 -4x +1 =0


Hot goIITian

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17 Apr 2012 00:34:58 IST
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 its discriminant is negative

 


Hot goIITian

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17 Apr 2012 00:38:02 IST
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the easiest way to calculate sin 18 approx iscalculate sin15 which is approx equal to sin 18sin(15+3)=sin15 cos3+ cos15 sin3sin3 =0 approxcos3=1 approxso calculate sin15

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12 Jun 2012 10:28:58 IST
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how to find the value of cos 18 degree?

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31 Jul 2013 12:23:19 IST
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1

 


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31 Jul 2013 12:24:36 IST
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1

 




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