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Решить тригонометрические уравнения, картинка прилагается
Решить тригонометрические уравнения, картинка прилагается
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20.9. cos x - cos 2x = sin 3xcos x - cos 2x = -2sin ((x+2x)/2) * sin ((x-2x)/2) = 2sin(3x/2)*sin(x/2)
sin 3x = 2sin(3x/2)*cos(3x/2)
Получаем
2sin(3x/2)*sin(x/2) = 2sin(3x/2)*cos(3x/2)
1) sin(3x/2) = 0; 3x/2 = pi*k; x1 = 2pi/3*k
2) sin(x/2) = cos(3x/2) = cos(x/2)*(4cos^2(x/2) - 3)
tg(x/2) = 4cos^2(x/2) - 3 = 4/(1 + tg^2 (x/2)) - 3
Замена tg(x/2) = t
t = 4 / (1 + t^2) - 3
t*(1 + t^2) = 4 - 3(1 + t^2)
t + t^3 = 4 - 3 - 3t^2 = 1 - 3t^2
t^3 + 3t^2 + t - 1 = 0
t^3 + t^2 + 2t^2 + 2t - t - 1 = (t + 1)(t^2 + 2t - 1) = 0
t1 = tg(x/2) = -1; x/2 = -pi/4 + pi*k; x2 = -pi/2 + 2pi*k
t^2 + 2t - 1 = (t + 1 - √2)(t + 1 + √2) = 0
t1 = tg(x/2) = -√2 - 1; x3 = -2arctg(√2 + 1) + pi*n
t2 = tg(x/2) = √2 - 1; x4 = 2arctg(√2 - 1) + pi*m
Ответ: x1 = 2pi/3*s; x2 = -pi/2 + 2pi*k;
x3 = -2arctg(√2 + 1) + pi*n; x4 = 2arctg(√2 - 1) + pi*m
20.10. cos 5x + cos 7x = cos(pi + 6x) = -cos 6x
2cos 6x*cos x = -cos 6x
1) cos 6x = 0;
6x1 = pi/2 + 2pi*k; x1 = pi/12 + pi/3*k
6x2 = 3pi/2 + 2pi*n; x2 = pi/4 + pi/3*n
2) 2cos x = -1; cos x = -1/2;
x3 = 2pi/3 + 2pi*m
x4 = 4pi/3 + 2pi*s