1) sinx = -1/2;
x = (-1)^(n+1)* arcsin(|-1/2|) + pi*n;
x = (-1)^(n+1)* pi/6) + pi*n; n ∈ Z
n = 0; x = -pi/6 ∉[0;3p]
n = 1; x = pi/6 + pi = 7pi/6 ∈<span>[0;3p]
</span>n = 2; x = -pi/6 + 2pi = 11pi/6 ∈<span>[0;3p]
</span>n = 3; x = pi/6 + 3pi ∉<span>[0;3p]
</span>Ответ: x = 7pi/6 ∪ x = 11pi/6
2) sinx = 1/2;
x = (-1)^(n)* arcsin1/2) + pi*n;
x = (-1)^(n)* pi/6)+ pi*n; n ∈ Z
n = -1; x = -pi/6 - pi ∉ [-p/2;3p/2]
n = 0; x = pi/6 ∈[-p/2;3p/2]
n = 1; x = -pi/6 + pi = 5pi/6 ∈[-p/2;3p/2]
n = 2; x = pi/6 + 2pi ∉[-p/2;3p/2]
Ответ: x = pi/6 ∪ x = 5pi/6
3) sinx = -√2/2;
x = (-1)^(n+1)* arcsin(|-√2/2|) + pi*n;
x = (-1)^(n+1)* pi/4) + pi*n; n ∈ Z
n = -4; x = -pi/4 - 4pi ∉[-3p;0]
n = -3; x = pi/4 - 3pi = -11pi/4 ∈[-3p;0]
n = -2; x = -pi/4 -2pi = -9pi/4 ∈[-3p;0]
n = -1; x = pi/4 - pi = - 3pi/4 ∈[-3p;0]
n = 0; x = -pi/4 ∈[-3p;0]
n = 1; x = pi/4 + pi ∉[-3p;0]
Ответ: x = -11pi/4 ∪ x = -9pi/4 ∪ x = pi/4 - pi ∪ x = -pi/4
4) sinx = √2/2;
x = (-1)^(n)* arcsin(√2/2) + pi*n;
x = (-1)^(n)* pi/4)+ pi*n; n ∈ Z
n = -2; x = pi/4 - 2pi = -7pi/4 ∉[-3p/2;5p/2]
n = -1; x = -pi/4 - pi = - 5pi/4 ∈[-3p/2;5p/2]
n = 0; x = pi/4 ∈[-3p/2;5p/2]
n = 1; x = -pi/4 + pi = 3pi/4 ∈<span>[-3p/2;5p/2]
</span>n = 2; x = pi/4 + 2pi = 9pi/4 ∈<span>[-3p/2;5p/2]
</span>n = 3; x = -pi/4 + 3pi ∉[-3p/2;5p/2]
Ответ: x = -5pi/4 ∪ x = pi/4 ∪ x = 3pi/4 ∪ x = 9pi/4
5) sinx = -√3/2;
x = (-1)^(n+1)* arcsin(|-√3/2|) + pi*n;
x = (-1)^(n+1)* pi/3) + pi*n; n ∈ Z
n = -2; x = -pi/3 - 2pi ∉[-2p;2p]
n = -1; x = pi/3 - pi = -2pi/3;
n = 0; x = -pi/3 ∈[-2p;2p]
n = 1; x = pi/3 + pi = 4pi/3 ∈[-2p;2p]
n = 2; x = -pi/3 + 2pi = 5pi/3 ∈[-2p;2p]
n = 3; x = pi/3 + 3pi ∉[-2p;2p]
Ответ: x = -2pi/3 ∪ x = -pi/3 ∪ x =4pi/3 ∪ x = 5pi/3
(x^2 + 7)^2 - 13(x^2 + 7) + 36 = 0
Пусть x^2 + 7 = t
t^2 - 13t + 36 = 0
D = 169 - 144 = 25
t1 = (13 + 5)/2 = 18/2 = 9
t2 = (13 - 5)/2 = 8/2 = 4
Обратная замена
x^2 + 7 = 9
x^2 = 2
x1 = - √2
x2 = √2
x^2 + 7 = 4
x^2 = - 3
нет решений
Ответ
± √2
a) (2b-5)²-36=((2b-5)-6)*((2b-5)+6)=(2b-11)*(2b+1)
б) 9-(7+3a)²= (3-(7+3a))*(3+(7+3a))=(3-7-3a)*(3+7+3a)=
=(-4-3a)*(10+3a)= - (4+3a)*(10+3a)
в) (4-11m)²-1=((4-11m)-1)*((4-11m)+1)=(4-11m-1)*(4-11m+1)=(3-11m)*(5-11m)
г) p²-(2p+1)²=(p-(2p+1))*(p+(2p+1)=(p-2p-1)*(p+2p+1)=(-p-1)*(3p+1) = -(p+1)*(3p+1)
Варианты д и е напишу в комментариях
X = 7-y^2
y^2 (7-y^2) = 12
7y^2 -y^4 = 12
y^4 - 7y^2 +12 = 0
Замена: a=y^2
a^2 - 7a - 12=0
a = 4
a=-3
по т. виета
y = 2 и -2
(из -3 не вычесть корень)
x* 4 = 12
x=3
ответ: (3;-2); (3;-2)
