$sin(x+30^\circ)cosy-cosxcos(y+60^\circ)-cos(30^\circ)sin(x+y)$

A cura di: Francesco Speciale

Semplificare la seguente espressione
$sin(x+30^circ)cosy-cosxcos(y+60^circ)-cos(30^circ)sin(x+y)$

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$sin(x+30^circ)cosy-cosxcos(y+60^circ)-cos(30^circ)sin(x+y)=$
Tenendo presente le formule di somma del seno e del coseno:

$sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)$
$cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)$

Riferendoci alla nosta espressione avremo:
$sin(x+30^circ)cosy-cosxcos(y+60^circ)-cos(30^circ)sin(x+y)=$
$(sinxcos(30^circ)+sin(30^circ)cosx)cosy-cosx(cosycos(60^circ)-sinysin(60^circ))-(sqrt3)/2(sinxcosy+cosxsiny)=$
$((sqrt3)/2sinx+1/2cosx)cosy-cosx(1/2cosy-(sqrt3)/2siny)-(sqrt3)/2sinxcosy-(sqrt3)/2sinycosx=$
$(sqrt3)/2sinxcosy+1/2cosxcosy-1/2cosycosx+(sqrt3)/2sinycosx-(sqrt3)/2sinxcosy-(sqrt3)/2sinycosx=$
Semplificando si ha che l’espressione
$sin(x+30^circ)cosy-cosxcos(y+60^circ)-cos(30^circ)sin(x+y)=0$.