Задание 9
Решите уравнение -
ответ:​

Решение
1)  2sin x-1=0
sinx = 1/2
x = (-1)^n arcsin(1/2) + πk, k∈Z
x = (-1)^n (π/6) + πk, k∈Z
2)  cos(2x+П/6)+1=0
cos(2x+П/6) = - 1
2x+П/6 = π + 2πn, n∈Z
2x = π - π/6 + 2πn, n∈Z
2x = 5π/6 + 2πn, n∈Z
x = 5π/12 + πn, n∈Z
3)  6sin²x - 5cosx + 5 = 0
6(1 - cos²x) - 5cosx + 5 = 0
6 - 6cos²x - 5cosx + 5 = 0
6cos²x + 5cosx - 11 = 0
cosx = t, ItI ≤ 1
6t² + 5t - 11 = 0
D = 25 + 4*6*11 = 289
t₁ = (- 5 - 17)/12
t₁ = - 22/12
t₁ = -11/6
t₁ = - 1 (5/6) не удовлетворяет условию ItI ≤ 1
t₂ = (- 5 + 11)/12
t₂ = 1/2
cosx = 1/2
x = (+ -)arccos(1/2) + 2πm, m∈Z
 x = (+ -) *(π/3) + 2πm, m∈Z

1) Сумма бесконечной убывающей геометрической прогрессии
S = b1/(1 - q)
У нас b1 = 8, q = 0,5, S = 8/(1 - 0,5) = 16
2) Арифметическая прогрессия
a(n) = a1 + d*(n - 1)
У нас a1 = 3, d = 4, n = 10, a(10) = 3 + 4*9 = 3 + 36 = 39
3) b1 = 9, q = -1/3, S = 9/(1 - 1/3) = 9/(2/3) = 9*3/2 = 13,5
4) Сумма арифметической прогрессии
S = (a1 + a(n))*n/2
a1 = 2, n = 102-2+1 = 101, a(101) = 102
S = (2 + 102)*101/2 = 52*101 = 5252
5) a1 = -3, d = -3, n = 25, a(25) = -3 - 3*24 = -3 - 72 = -75
6) a1 = 10, d = -2, n = 10, a(10) = 10 - 2*9 = 10 - 18 = -8
S(10) = (10 - 8)*10/2 = 2*10/2 = 10