oppgave
løs likningen:
sin(x- [symbol:pi] /4) pluss cos(x- [symbol:pi] /4) = 1
x e ( 0,2 [symbol:pi] )
svar = [symbol:pi] /4 eller 3 [symbol:pi] /4
oppgave:
skriv disse uttrykkene enklere.
a) cos(x pluss [symbol:pi] /3) pluss sin ( x pluss [symbol:pi] /6)
b) sin x/ sin(x pluss [symbol:pi] /6 - sin ( x- [symbol:pi] /6)
svaret på a= cos x og = tan x
skjønner ikke
Moderatorer: Aleks855, Gustav, Nebuchadnezzar, Janhaa, DennisChristensen, Emilga
a) cos(x+ [symbol:pi] /3) + sin(x + [symbol:pi] /6)
=
cos(x)cos( [symbol:pi] /3) - sin(x)sin([symbol:pi] /3) + sin(x)cos([symbol:pi] /6) + cos(x)sin([symbol:pi] /3)
=
cos(x) * 1/2 - sin(x) * 1/2* [symbol:rot] 3 + sin(x)* 1/2* [symbol:rot] 3 + cos(x) * 1/2
= cos(x) * 1/2 + cos(x) * 1/2 = 2*(1/2)*cos(x) = cos(x)
b) sin(x) / (sin(x + [symbol:pi] /6) - sin(x - [symbol:pi] /6)
= (skriver bare ut for nevneren)
sin(x)cos([symbol:pi] /6) + cos(x)sin([symbol:pi] /6) - (sin(x)cos([symbol:pi] /6) - cos(x)sin([symbol:pi] /6))
= sin(x) * 1/2* [symbol:rot] 3 + 1/2 * cos(x) - sin(x) * 1/2* [symbol:rot] 3 + 1/2 * cos(x)
= 2 * 1/2 * cos(x) = cos(x)
= sin(x)/cos(x) = tan(x)
du må bruke formlene for sum og differanse:
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
står mer her: http://en.wikipedia.org/wiki/Cosinus (under "Identities")
=
cos(x)cos( [symbol:pi] /3) - sin(x)sin([symbol:pi] /3) + sin(x)cos([symbol:pi] /6) + cos(x)sin([symbol:pi] /3)
=
cos(x) * 1/2 - sin(x) * 1/2* [symbol:rot] 3 + sin(x)* 1/2* [symbol:rot] 3 + cos(x) * 1/2
= cos(x) * 1/2 + cos(x) * 1/2 = 2*(1/2)*cos(x) = cos(x)
b) sin(x) / (sin(x + [symbol:pi] /6) - sin(x - [symbol:pi] /6)
= (skriver bare ut for nevneren)
sin(x)cos([symbol:pi] /6) + cos(x)sin([symbol:pi] /6) - (sin(x)cos([symbol:pi] /6) - cos(x)sin([symbol:pi] /6))
= sin(x) * 1/2* [symbol:rot] 3 + 1/2 * cos(x) - sin(x) * 1/2* [symbol:rot] 3 + 1/2 * cos(x)
= 2 * 1/2 * cos(x) = cos(x)
= sin(x)/cos(x) = tan(x)
du må bruke formlene for sum og differanse:
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
står mer her: http://en.wikipedia.org/wiki/Cosinus (under "Identities")