Question

1.In reperul cartezian xoy se considera punctele A(3,2),B(-1,4),C(5,4).Determinati coordonatele punctului M astfel incat AM vector=3AB vector-2AC vector.
2.Determinați x apartine [0,pi] pentru care cos2x=sin (x+pi pe 2)
Rapid!!!!29 de puncte + coronița

Answer 1

AM vector=3AB vector-2AC vector.
 in continuare, toti sunt vectori;

AO+OM=3(AO+OB)-2(AO+OC)
AO+OM=AO+3OB-2OC
OM=3OB-2OC
OM=3(-i+4j)-2(5i+4j)=-3i+12j-10i-8j=-13i+4j⇔M(-13;4)


sin (x+π/2)=sinxcoπ/2+sinπ/28cosx=sinx*0+1*cosx=cosx
cos2x=cosx
2cos²x-1=cosx
cosx=t
2t²-t-1=0
rezolvand cuΔ, obtinem
t1=-1/2 si t2=1

t1=cosx=-1/2
solutiiin R
x=2kπ+-arccos(-1/2)=2kπ+-(2π/3)
convine doar 2π/3∈(0;π)

t2=1
x=2kπ+-arccos1=2kπ+-π
convine x=0
deci S= {0;2π/3}

verificare cos0=sinπ/2=1; adevarat
cos4π/3=sin7π/6
cos120°=sin 210° =-1/2 adevarat