А)sin(x-П/3) * cos(х-П/6)= 1 б)sin x/2 * sin 3x/2= 1/2 в) 2sin (П\4+х) * sin (П/4-х) + sin^2x=0

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вопрос

а) sin(x-π/3) cos(x-π/6) = 1
sin(x)cos(π/3)cos(x)sin(π/6) = 1
sin(x) (sqrt(3)/2) cos(x) (1/2) = 1
(sin(x) sqrt(3) cos(x)) / 2 = 1
(sin(x) sqrt(3) cos(x)) = 2
sin(2x)sqrt(3) = 2
(2sqrt(3)/2) = 2
sqrt(3) = 1
This equation is false, so the initial statement is not true.

б) sin(x/2) sin(3x/2) = 1/2
(sin(x)cos(x)/2) (3sin(x)cos(x)/2) = 1/2
(3sin^2(x)cos^2(x))/4 = 1/2
(3sin^2(x)(1-sin^2(x)))/4 = 1/2
(3sin^2(x) - 3sin^4(x))/4 = 1/2
3sin^2(x) - 3sin^4(x) = 2
3sin^2(x) - 3(1 - cos^2(x))^2 = (2)
3sin^2(x) - 3(1 - cos(x))^2 = (2)
6sin^2(x) - 3 = 2
6sin^2(x) = 5
sin^2(x) = 5/6
sin(x) = sqrt(5/6)
This equation is correct.

в) 2sin(π/4 + x) sin(π/4 - x) + sin^2(x) = 0
2(sin(π/4)cos(x) + cos(π/4)sin(x)) (sin(π/4)cos(x) - cos(π/4)sin(x)) + sin^2(x) = 0
2((sqrt(2)/2)cos(x) + (sqrt(2)/2)sin(x)) ((sqrt(2)/2)cos(x) - (sqrt(2)/2)sin(x)) + sin^2(x) = 0
2(sqrt(2)(cos^2(x) - sin^2(x))/2) + sin^2(x) = 0
2(sqrt(2)(1 - 2sin^2(x))) + sin^2(x) = 0
2sqrt(2) - 4sqrt(2)sin^2(x) + sin^2(x) = 0
(2sqrt(2) - 3sin^2(x)) = 0
3sin^2(x) = 2sqrt(2)
sin^2(x) = 2sqrt(2)/3
sin(x) = sqrt(2/3 sqrt(2))
This equation is correct.