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Trignometry
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Let A = 18^{0}
=> 5A = 90^{0}
=> 3A + 2 A = 90^{0}
=> 3A = 90^{2}  2A
=> Sin 3A = Cos 2A
=> 3 Sin A  4 Sin^{3} A = 1  2 Sin^{2} A
=> 3 Sin A  4 Sin^{3} A  1 + 2 Sin^{2} A = 0
=> 4 Sin^{3}A  2 Sin^{2} A  3 Sin A + 1 = 0
putting sin A = x we have,
4 x^{3}  2 x^{2}  3x + 1 = 0
=> 4x^{3}  4 x^{2} + 2 x^{2}  2x  x + 1 = 0
=> 4 x^{2}(x  1) + 2x (x  1)  1 (x  1) = 0
=> (x  1) (4x^{2} + 2x  1) = 0
either, x = 1 or 4x^{2} + 2x  1 = 0
But, since Sin 18^{0} cannot be equal to 1 so,
4x^{2} + 2x  1 = 0
Solve this quadratic eqn. and you will get the value of sin 18^{0} i.e., root (5)  1 / 4
Then find cos x
Then Sin 36^{0} = 2 sin 18<sup>0</sup> . cos 18<sup>0</sup>
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