3
a)x+y=π/2⇒x=π/2-y
sin²x-sin²y=1⇒sin²(π/2-y)-sin²y=1⇒cos²y-sin²y=1⇒cos2y=1⇒2y=0⇒y=0
x=π/2-0=π/2
b)x-y=π/6⇒x=y+π/6
sinx*cosy=1/2⇒sin(y+π/6)*siny=1/2⇒1/2(sinπ/6+sin(2y+π/6))=
=1/2⇒1/2+sin(2y+π/6)=1⇒sin(2y+π/6)=1/2⇒2y+π/6=π/6⇒2y=0⇒y=0
x=0+π/6=π/6
4
a)sin4x-sinx=0
2sin(3x/2)cos(5x/2)=0
sin(3x/2)=0⇒3x/2=πn⇒x=2πn/3 ⇒x=8π/3∈[3π;5π/2]
cos(5x/2)=0⇒5x/2=π/2+πn⇒x=π/5+2πn/5 ⇒x=11π/5∈[3π;5π/2]
b)2sin(π/2-x)*cos(π/2+x)=√3cosx
2cosx*(-sinx)=√3cosx
√3cosx+2cosxsinx=0
cosx(√3+2sinx)=0
cosx=0⇒x=π/2+πn ⇒x={-3π/2;-π/2}∈[-2π;-π/2]
sinx=-√3/2⇒x=(-1)^n+1*π/3+πn x =-2π/3∈[-2π;-π/2]
1)x^2-16x+64>=0
D=(-16)^2-4*64=0
X=16/2=8
x^2-16x+64=(х-8)^2
(х-8)^2>=0
Перед 8 есть цифра 7 ,так проверим( 7-8)^2=1 значит +
После 8 есть цифра 10, (10-8)^2=4 значит +
И ответ ( -бесконечность;8]U[8;+бесконечность)
Sₙ = n(2a₁+d(n-1)) / 2
S₁₅ - S₁₄ = 15(2a₁+14d) / 2 - 14(2a₁+13d) / 2 = 87
2a₁+ 14(15-13)d = 87*2
a₁ + 14d = 87
S₁₁ - S₁₀ = 11(2a₁+10d) / 2 - 10(2a₁+ 9d) / 2 = 43
2a₁ + 10(11-9)d = 43*2
a₁ + 10d = 43
14d - 10d = 87 - 43 = 44
d = 11