Get the answer to "What is the value of sin2(π10)sin2(3π10)?"

Gavin Weiss

Answered question

2022-03-13

What is the value of

\sin^{2} (\frac{π}{10}) \sin^{2} (\frac{3 π}{10})

elxanaijm

Beginner2022-03-14Added 5 answers

\sin^{2} {\frac{π}{10}} \sin^{2} \frac{3 π}{10} = {(\frac{4 \sin {\frac{π}{5}} \cos {\frac{2 π}{10}} \cos {\frac{4 π}{10}}}{4 \sin \frac{π}{5}})}^{2} =

= {(\frac{\sin \frac{4 π}{5}}{4 \sin \frac{π}{5}})}^{2} =

= \frac{1}{16}

Kasen Alexander

Beginner2022-03-15Added 7 answers

The roots of

Φ_{10} (x) = 1 - x + x^{2} - x^{3} + x^{4}

are the primitive tenth roots of unity,

ξ, ξ^{3}, ξ^{7}, ξ^{9}

with

ξ = \exp (\frac{2 π i}{10})

. The roots of

Φ_{20} (x) = Φ_{10} (x^{2}) = 1 - x^{2} + x^{4} - x^{6} + x^{8}

are the primitive 20-th roots of unity,

ζ, ζ^{3}, ζ^{7}, ζ^{9}, ζ^{11}, ζ^{13}, ζ^{17}, ζ^{19}

with

ζ = \exp (\frac{π i}{10})

. By dividing

Φ_{20} (x)

x^{4}

and by writing what we get as a polynomial in

(x + \frac{1}{x})

we get that

x^{4} - \frac{5}{4} x^{2} + \frac{5}{16} = \prod_{k \in {1, 3, 7, 9}} (x - \cos \frac{π k}{10})

and by evaluating both sides at x=-1 we get:

\prod_{k \in {1, 3, 7, 9}} (1 + \cos \frac{π k}{10}) = 1 - \frac{5}{4} + \frac{5}{16} = \frac{1}{16}

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