Question

Aratati ca ecuatia $x^{3}-3x+1=0$ ,are solutia x=2cos40°

Answer 1

x³-3x+1=0⇔x(x²-3)=-1
x²-3=4cos²40-3=4·(1+cos80)/2-3=2(1+cos80)-3=2cos80-1
Atunci:
x(x²-3)=2cos40(2cos80-1)=4cos80cos40-2cos40=
=4·1/2[cos120+cos40]-2cos40=2(-1/2+cos40)-2cos40=
-1+2cos40-2cos40=-1 ,
ceea ce trebuia demonstrat.
Deci x=2cos40 este solutie a ecuatiei.
OBS.
1)cos120=cos(180-60)=-cos60=-1/2
2) am folosit: cos²x=(1+cos2x)/2
3)cosp·cosq=1/2[cos(p+q)+cos(p-q)]