Question

Calculati sin pi /10 plsss

Answer 1

sin(pi/10) = sin(18)

Notam x = 18.

x = 18

5x = 90

2x+3x = 90

2x = 90-3x

sin(2x) = sin(90-3x)

sin(2x) = cos(3x)

2sinxcosx = 4cos³x-3cosx |:cos(x)

2sinx = 4cos²x-3

2sinx = 4(1-sin²x)-3

Notăm sinx = t

=> 2t = 4(1-t²)-3

=> 2t = 4-4t²-3

=> 4t²+2t-1 = 0

Delta = 4+16 = 20

=> t₁,₂ = (-2±√20)/8

Dar sin(18) > 0

=> t = (-2+√20)/8 = (-2+2√5)/8 = (√5-1)/4

=> sin(18) = (√5-1)/4

=> sin(pi/10) = (√5-1)/4