@capedsam Tweeted: “Can you find an exact value for cos(pi/5)? “

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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:
x

………(1)
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We raise both sides of equation (1) to the

power:
x

……………………….(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:
x

 ……..(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:
x

……………………………..(4)
x

……………………………….(5)
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We can substitute

in (5) to get:
x

………….(6)
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Expanding (6) and dividing through by

we get:
x

………………………………………(7)
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This is a quadratic equation for

with roots

x
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:
x

.
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