Posted by: Gary Ernest Davis on: December 3, 2010

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It is actually fairly easy to find an exact value for by following a path through the complex numbers.

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To do this we use Euler’s identity for all real numbers .

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Jim Tanton (@jamestanton) has a very nice introduction to Euler’s identity here.

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As shorthand let’s write .

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From Euler’s identity we know that:

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………(1)

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We raise both sides of equation (1) to the power:

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……………………….(2)

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The left hand side of equation (2) is .

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The right hand side of equation (2) is, by the binomial theorem:

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Â ……..(3)

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We separate the expression in (3) into real and imaginary parts, and equate the real part to -1, and the imaginary part to 0:

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……………………………..(4)

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……………………………….(5)

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We can substitute in (5) to get:

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………….(6)

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Expanding (6) and dividing through by we get:

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………………………………………(7)

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This is a quadratic equation for with roots

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This gives

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Only one of these two roots can be equal to .

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Which one?

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We know so .

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Therefore, so which means, since , that:

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.

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One can also get the quadratic equation for from the multiple angle formulas (if one can remember them!).

1 | dgradcliffe

December 3, 2010 at 11:17 am

Note also that cos(pi/5) = (1+sqrt(5))/4, which is half the golden ratio. Can you find a geometric proof?

Gary Davis

December 3, 2010 at 11:21 am

Nice question. Thanks.

Of course

@capedsam

January 14, 2011 at 12:29 am

1). Connect all of the vertices of a regular pentagon.

2). Use similar triangle ratios to show that the sides of the 36-72-72 isosceles triangle are related by the golden ratio.

3). Use law of cosines to prove that cos(36) = phi/2