Помогите с задачей по математике.
1. Решите уравнения.
а) −6cos(x)+3√3=0−6cos(x)+33=0;
б) sin(x3+π3)=−1sin(x3+π3)=−1;
в) 2sin2(x)−9cos(x)−6=02sin2(x)−9cos(x)−6=0;
г) 6sin2(x)−7sin(x)cos(x)+7cos2(x)=06sin2(x)−7sin(x)cos(x)+7cos2(x)=0.
2. Решите уравнение 5sin2(x)−5sin(x)cos(x)−2cos2(x)=−15sin2(x)−5sin(x)cos(x)−2cos2(x)=−1.
а) −6cos(x) + 3√3 = 0
cos(x) = 3√3/6 = √3/2
x = +- π/6 + 2π*k, k принадлежит Z
б) sin(x/3+π/3) = −1
x/3+π/3 = 3π/2 + 2π*k
x/3 = 3π/2 - π/3 + 2π*k = 9π/6 - 2π/6 + 2π*k = 7π/6 + 2π*k
x = 7π/2 + 6π*k, k принадлежит Z
в) 2sin^2 (x) − 9cos(x) − 6 = 0
2(1 - cos^2 (x)) - 9cos(x) - 6 = 0
2cos^2 (x) + 9cos(x) + 4 = 0
D = 9^2 - 4*2*4 = 81 - 32 = 49 = 7^2
cos(x1) = (-9-7)/4 = -16/4 = -4 - нет решений
cos(x2) = (-9+7)/4 = -2/4 = -1/2
x = +- 2π/3 + 2π*k, k принадлежит Z
г) 6sin^2 (x) − 7sin(x)cos(x) + 7cos^2 (x) = 0.
Делим всё на cos^2 (x)
6tg^2 (x) - 7tg(x) + 7 = 0
D = 7^2 - 4*6*7 = 49 - 168 < 0
Решений нет
2. Решите уравнение
5sin^2 (x) − 5sin(x)cos(x) − 2cos^2 (x) = −1.
5sin^2 (x) − 5sin(x)cos(x) − 2cos^2 (x) = - sin^2 (x) - cos^2 (x)
6sin^2 (x) − 5sin(x)cos(x) − cos^2 (x) = 0
Делим всё на cos^2 (x)
6tg^2 (x) - 5tg(x) - 1 = 0
D = 5^2 - 4*6(-1) = 25 + 24 = 49 = 7^2
tg(x1) = (5-7)/12 = -2/12 = -1/6
x1 = -arctg(1/6) + π*k, k принадлежит Z
tg(x2) = (5+7)/12 = 1
x2 = π/4 + π*k, k принадлежит Z
а) 3 sqrt(3) - 6 cos(x) = 33 - 6 cos(x) = 0
б) -sin(x^3) = sin(x^3) = -1
