3)cos^2 x-3sin x*cos x = -1 cos^2 x - 3sin x*cos x + 1 = 0 cos^2 x - 3sin x*cos x + cos^2 x + sin^2 x = 0 sin^2 x - 3sin x*cos x + 2cos^2 x = 0 Делим все на cos^2 x tg^2 x - 3tg x + 2 = 0 (tg x - 1)(tg x - 2) = 0 tg x = 1; x1 = pi/4 + pi*k tg x = 2; x2 = arctg(2) + pi*n
sin 3x = sin(x + 2x) = sin x*cos 2x + cos x*sin 2x =
= sin x*(1 - 2sin^2 x) + cos x*2sin x*cos x = sin x*(1 - 2sin^2 x + 2cos^2 x) =
= sin x*(1 - 2sin^2 x + 2 - 2sin^2 x) = sin x*(3 - 4sin^2 x)
cos 2x = 1 - 2sin^2 x
Подставляем
sin x*(3 - 4sin^2 x) - √3(1 - 2sin^2 x) - sin x = 0
Замена sin x = t и раскрываем скобки
3t - 4t^3 - √3 + 2√3*t^2 - t = 0
Умножаем все на -1, чтобы старший член был положительным
4t^3 - 2√3*t^2 - 2t + √3 = 0
2t^2*(2t - √3) - (2t - √3) = 0
(2t - √3)(2t^2 - 1) = 0
(2t - √3)(t√2 - 1)(t√2 + 1) = 0
t1 = sin x = √3/2; x1 = pi/3 + 2pi*k; x2 = 2pi/3 + 2pi*k
t2 = sin x = -1/√2; x3 = 5pi/4 + 2pi*n; x4 = 7pi/4 + 2pi*n
t3 = sin x = 1/√2; x5 = pi/4 + 2pi*m; x6 = 3pi/4 + 2pi*m
Корни x3, x4, x5, x6 можно объединить в один:
x3 = pi/4 + pi/2*n
ответ: x1 = pi/3 + 2pi*k; x2 = 2pi/3 + 2pi*k; x3 = pi/4 + pi/2*n
2)2sin(40+x)*sin(x-50) + 1 = 0
2sin(90+x-50)*sin(x-50) = -1
2cos(x-50)*sin(x-50) = sin(2x-100) = -1
2x - 100 = 270 + 360*n
x = 185 + 180*n = 5 + 180*k
3)cos^2 x-3sin x*cos x = -1
cos^2 x - 3sin x*cos x + 1 = 0
cos^2 x - 3sin x*cos x + cos^2 x + sin^2 x = 0
sin^2 x - 3sin x*cos x + 2cos^2 x = 0
Делим все на cos^2 x
tg^2 x - 3tg x + 2 = 0
(tg x - 1)(tg x - 2) = 0
tg x = 1; x1 = pi/4 + pi*k
tg x = 2; x2 = arctg(2) + pi*n
1) 25a^2 - c^2
2) 9x^2 - 16y^2
3) x^2 - 4y^2
4) x^3 - 8y^3
5) 27a^3 - 64b^3
6) 8x^3 - 125y^3
7) a^3 - a^2 b + ab^2 - b^3
8) x^2 - b^2 - ax -ab
9) 3b + bc + 3ac + 9a
10) a^2 x^2 - y^4
11) a^2 y^2 - x^6
12) c^2 - 4c + 4 - 9x^2
13) c^2 - 6c + 9 -4x^2
14) 4c^2 + 20c + 25 - 9a^2
15) y^2 x + y + y x^2 + x + 4yx +4
16) 3x^2 + 2x - xy - 2y^2 + y^3 - 3xy^2
17) x^2 + x - xy - y^2 + y^3 - xy^2
18) a^2 x + a +a x^2 + x + 2ax + 21) 25a^2 - c^2 = (5a+c)(5a-c)
2) 9x^2 - 16y^2 = (3x-4y)(3x+4y)
3) x^2 - 4y^2 = (x-2y)(x+2y)
4) x^3 - 8y^3 =(x-2y)(x^2+4y^2+2xy)
5) 27a^3 - 64b^3 =(3a-4b)(9a^2+16b^2+12ab)
6) 8x^3 - 125y^3 =(2x-5y)(4x^2+25y^2+10xy)
7) a^3 - a^2 b + ab^2 - b^3 =a^2(a-b)+b^2(a-b)=(a^2+b^2)(a-b)
8) x^2 - b^2 - ax -ab =(x-b)(x+b)-a(x+b)=(x+b)(x-b-a)
9) 3b + bc + 3ac + 9a = b(3+c)+3a(c+3)=(3+c)(3a+b)
10) a^2 x^2 - y^4 =(ax-y^2)(ax+y^2)
11) a^2 y^2 - x^6 =(ay-x^3)(ay+x^3)
12) c^2 - 4c + 4 - 9x^2
13) c^2 - 6c + 9 -4x^2 =(c-3)^2-4x^2=(c-3-2x)(x-3+2x)
14) 4c^2 + 20c + 25 - 9a^2= (2c+5)^2-9a^2=(2c+5-3a)(2c+5+3a)
15) y^2 x + y + y x^2 + x + 4yx +4
16) 3x^2 + 2x - xy - 2y^2 + y^3 - 3xy^2
17) x^2 + x - xy - y^2 + y^3 - xy^2
18) a^2 x + a +a x^2 + x + 2ax + 2=