Trigonometria

  • Materia: Trigonometria
  • Visto: 1390
  • Data: 09/01/2010
  • Di: Redazione StudentVille.it

$((cos^2(18^\\circ))-(sin^2(18^\\circ)))/((cos^2(240^\\circ)))-(tg^2(-18^\\circ))$

esercizio svolto o teoria

A cura di: Francesco Speciale

Calcolare il valore della seguente espressione:
$((cos^2(18^\circ))-(sin^2(18^\circ)))/((cos^2(240^\circ)))-(tg^2(-18^\circ))$



$(cos^2(18^\circ)-sin^2(18^\circ))/(cos^2(240^\circ))-tg^2(-18^\circ)=
Essendo $cos(18^\circ)=1/4sqrt(10+2sqrt5) , sin(18^\circ)=1/4(sqrt5-1) , cos(240^\circ)=-1/2 , tg(-18^\circ)=-sqrt(1-2/5sqrt5)$,
sostituendo nell'espressione si ha:
$=((1/4sqrt(10+2sqrt5))^2-(1/4(sqrt5-1))^2)/((-1/2)^2)-(-sqrt(1-2/5sqrt5))^2=$
$=(1/(16)(10+2sqrt5)-1/(16)(sqrt5-1)^2)/(1/4)-(1-2/5sqrt5)=$
$=(1/(16)(10+2sqrt5)-1/(16)(5+1-2sqrt5))/(1/4)-(1-2/5sqrt5)=$
$=(5/8+(sqrt5)/8-3/8+(sqrt5)/8)*4-1+2/5sqrt5=(1/4+(sqrt5)/4)*4-1+2/5sqrt5=$
$=1+sqrt5-1+2/5sqrt5=(5sqrt5+2sqrt5)/5=7/5sqrt5$.