matematikk.net

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

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")