(sinx+cosx)^2 -3(sinx+cosx) +2=0 Количество корен. на отрезке (0;2п) включая
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sedinalana
Sinx+cosx=a a²-3a+2=0 a1+a2=3 U a1*a2=2 a1=1⇒sinx+cosx=1 sinx+sin(π/2-x)=1 2sinπ/4cos(x-π/4)=1 √2cos(x-π/4)=1 cos(x-π/4)=1/√2 x-π/4=+-π/4+2πk x=π/4-π/4+2πk=2πk U x=π/4+π/4+2πk=π/2+2πk 0≤2πk≤π U 0≤π/2+2πk≤π 0≤k≤1/2 нет решения U 0≤1+4k≤2⇒-1≤4k≤1⇒-1/4≤k≤1/4 k=0⇒x=π/2 a2=2⇒sinx+cosx=2 √2cos(x-π/4)=2 cos(x-π/4)=√2>1 нет решения Ответ один корень
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a²-3a+2=0
a1+a2=3 U a1*a2=2
a1=1⇒sinx+cosx=1
sinx+sin(π/2-x)=1
2sinπ/4cos(x-π/4)=1
√2cos(x-π/4)=1
cos(x-π/4)=1/√2
x-π/4=+-π/4+2πk
x=π/4-π/4+2πk=2πk U x=π/4+π/4+2πk=π/2+2πk
0≤2πk≤π U 0≤π/2+2πk≤π
0≤k≤1/2 нет решения U 0≤1+4k≤2⇒-1≤4k≤1⇒-1/4≤k≤1/4 k=0⇒x=π/2
a2=2⇒sinx+cosx=2
√2cos(x-π/4)=2
cos(x-π/4)=√2>1 нет решения
Ответ один корень