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Empma u3 heptagono regular

Page 1

𝑧7 = 1 2πœ‹ 2πœ‹/7

𝑧7 = 1

𝑧 7 βˆ’ 1 = 0.

𝑧 7 βˆ’ 1 = (𝑧 βˆ’ 1)(𝑧 6 + 𝑧 5 + 𝑧 4 + 𝑧 3 + 𝑧 2 + 𝑧 + 1) = 0

𝑧=1 𝑧6 + 𝑧5 + 𝑧4 + 𝑧3 + 𝑧2 + 𝑧 + 1 = 0 πœƒ = 2πœ‹/7


2πœ‹

π‘§π‘˜ = 𝑒 π‘˜ 7 𝑖

π‘π‘Žπ‘Ÿπ‘Ž

2πœ‹

π‘§π‘˜ = 𝑒 π‘˜ 7 𝑖 = cos (π‘˜

π‘˜ = 0, 1, … ,6

2πœ‹ )+ 7

𝑖 βˆ™ 𝑠𝑒𝑛 (π‘˜

2πœ‹ ) 7

π‘˜=1

2πœ‹ 2πœ‹ 2πœ‹ 𝑧1 = 𝑒 7 𝑖 = cos ( ) + 𝑖 βˆ™ 𝑠𝑒𝑛 ( ) 7 7

2πœ‹

cos ( 7 )

𝑧1

2πœ‹

𝑧̅1 = 𝑒 βˆ’π‘–π‘˜ 7 = cos (

2πœ‹ 2πœ‹ ) βˆ’ 𝑖 βˆ™ 𝑠𝑒𝑛 ( ) 7 7

𝑧6 + 𝑧5 + 𝑧4 + 𝑧3 + 𝑧2 + 𝑧 + 1 = 0

2πœ‹

2πœ‹

𝑧1 + 𝑧̅1 = 𝑒 π‘–π‘˜ 7 + 𝑒 βˆ’π‘–π‘˜ 7 = 𝑧1 +

1 𝑧1

2πœ‹ 7

= 2 cos ( )


𝑧6 + 𝑧5 + 𝑧4 + 𝑧3 + 𝑧2 + 𝑧 + 1 = 0

𝑧3

1 𝑧

𝑧3 + 𝑧2 + 𝑧 + 1 + +

𝑧3 +

1 𝑧2

+

1 𝑧3

=0

1 1 1 + 𝑧2 + 2 + 𝑧 + +1=0 3 𝑧 𝑧 𝑧

1 3 3 1 2 1 (𝑧 + ) βˆ’ 3𝑧 βˆ’ + (𝑧 + ) βˆ’ 2 + ( 𝑧 + ) + 1 = 𝑧 𝑧 𝑧 𝑧 1 3 1 1 2 1 (𝑧 + ) βˆ’ 3 (𝑧 + ) + (𝑧 + ) + ( 𝑧 + ) βˆ’ 1 = 0 𝑧 𝑧 𝑧 𝑧 1 3

1 2

1

(𝑧 + 𝑧) βˆ’ 2 (𝑧 + 𝑧) + (𝑧 + 𝑧) βˆ’ 1 = 0

1

π‘₯ = 𝑧+𝑧

π‘₯ 3 + π‘₯ 2 βˆ’ 2π‘₯ βˆ’ 1 = 0

2πœ‹

2 cos ( 7 )

2πœ‹ 7

2cos( ) 2πœ‹ 7

cos( )


πœƒ=

2πœ‹ 9


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