A geometry problem by A Former Brilliant Member

Geometry Level 3

$\large \sin \left(\frac{2\pi}{7}\right) + \sin \left(\frac{4\pi}{7}\right) + \sin \left(\frac{8\pi}{7}\right)$

Find the closed form of the expression above to 3 decimal places.

1 solution

Prakhar Bindal
Apr 15, 2017

Well here u go

Call pi/7 = x

=>sin2x+sin4x+sin8x

=> sin2x+2sin6xcos2x

Notice sin2x = 2sinxcosx and sin6x = sinx

=> 2sinxcosx+2sinxcos2x

=> 2sinx(cosx+cos2x)

=>4sinxcos(3x/2)cos(x/2)

Changing everything to sin u get

=>4sinxsin2xsin3x

Now use infamous identity that

sin(pi/n)*sin(2pi/n)...........sin((n-1)pi/n) = n/2^(n-1)

Doing this sinxsin2xsin3xsin4xsin5xsin6x = 7/64

But sinx = sin6x , sin2x = sin5x , sin4x = sin3x

Hence sinxsin2xsin3x = root(7)/8

Hence required sum is 4*root(7)/8 = root(7)/2 = 1.32

thnk u bhai wo dusra bi dekiyo :)

A Former Brilliant Member - 4 years, 1 month ago

how to know so many identities ?

A Former Brilliant Member - 4 years, 1 month ago

Trigo is all about identities

Prakhar Bindal - 4 years, 1 month ago

dusra deka ? thnx for this one buddy :) bhai maine kisi oor ka farm ni bara , iit nikal jayegi na mujse :?

A Former Brilliant Member - 4 years, 1 month ago

one can do by complex number also assuming x^7-1=0

Ashutosh Sharma - 3 years, 2 months ago