By using euler formula,one can obtain: \(\displaystyle{2}{\sin{{\left({\frac{{\pi}}{{{180}}}}\right)}}}={\left({\left(-{1}\right)}^{{{\frac{{{1}}}{{{180}}}}}}\right)}^{{{89}}}-{\left({\left(-{1}\right)}^{{{\frac{{{1}}}{{{180}}}}}}\right)}^{{{91}}}\)

WesDiectstemiwxg

WesDiectstemiwxg

Answered question

2022-03-19

By using euler formula,one can obtain:
2sin(π180)=((1)1180)89((1)1180)91

Answer & Explanation

blindhaedzgs

blindhaedzgs

Beginner2022-03-20Added 2 answers

Let's try the following approach:
1=eπi+2kπi=eπi(2k+1), kZ(1)1180=eπt180(2k+1)
and from here
((1)1180)89((1)1180)91=(eπt180(2k+1))89(eπt180(2k+1))91
=e89πt180(2k+1)(1eπt90(2k+1))
Taking the real part of the above we get
cos89π(2k+1)180(1cosπ(2k+1)90)+sin89π(2k+1)180sinπ(2k+1)90
=cos89π(2k+1)180cos91π(2k+1)180=2sinπ(2k+1)sin2π(2k+1)90=0
The equality one before the last above follows from the trigonometric identity
cosxcosy=2sinx+y2sinxy2

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