№4
66 = 2 ·3 · 11
р₁=2
р₂=3
р₃=11
Находим по формуле (р₁+1)(р₂+1)(р₃+1) сумму всех делителей числа 66.
(2+1)·(3+1)·(11+1) = 3 · 4 · 12 = 144
Ответ: 144
№5
cos α = √91/10; 0° < α < 90°
sinα - ?
Воспользуемся формулой основного тригонометрического тождества
sin²α + cos²α = 1
Из неё выразим sinα через cosα.
sin²α = 1- cos²α
sin²α = 1 - (√91/10)² = 1 - 91/100 = 9/100
sin²α =9/100
При условии, что 0° < α < 90° значение sinα>0.
sinα = √(9/100) = 3/10 = 0,3
Ответ: sinα = 0,3
2Cos²x + 5Sinx + 1 = 0
2(1 - Sin²x) + 5Sinx + 1 = 0
2 - 2Sin²x + 5Sinx + 1 = 0
- 2Sin²x + 5Sinx + 3 = 0
2Sin²x - 5Sinx - 3 = 0
Сделаем замену : Sinx = m , где - 1 ≤ m ≤ 1
2m² - 5m - 3 = 0
D = (-5)² - 4 * 2 * (- 3) = 25 + 24 = 49 = 7²
![m_{1}=\frac{5-7}{4}=-\frac{1}{2}\\\\m_{2}=\frac{5+7}{4}=3](https://tex.z-dn.net/?f=m_%7B1%7D%3D%5Cfrac%7B5-7%7D%7B4%7D%3D-%5Cfrac%7B1%7D%7B2%7D%5C%5C%5C%5Cm_%7B2%7D%3D%5Cfrac%7B5%2B7%7D%7B4%7D%3D3)
Sinx = 3 - решений нет
![Sinx=-\frac{1}{2}\\\\1)x=arcSin(-\frac{1}{2})+2\pi n,n\in z\\\\x=-\frac{\pi }{6}+2\pi n,n\in z\\\\2)x=-\pi+\frac{\pi }{6}+2\pi n,n\in z\\\\x=-\frac{5\pi }{6}+2\pi n,n\in z](https://tex.z-dn.net/?f=Sinx%3D-%5Cfrac%7B1%7D%7B2%7D%5C%5C%5C%5C1%29x%3DarcSin%28-%5Cfrac%7B1%7D%7B2%7D%29%2B2%5Cpi+n%2Cn%5Cin+z%5C%5C%5C%5Cx%3D-%5Cfrac%7B%5Cpi+%7D%7B6%7D%2B2%5Cpi+n%2Cn%5Cin+z%5C%5C%5C%5C2%29x%3D-%5Cpi%2B%5Cfrac%7B%5Cpi+%7D%7B6%7D%2B2%5Cpi+n%2Cn%5Cin+z%5C%5C%5C%5Cx%3D-%5Cfrac%7B5%5Cpi+%7D%7B6%7D%2B2%5Cpi+n%2Cn%5Cin+z)
Отрезок не записан.
Если отрезок π < x < 2π , то корни :
![\frac{7\pi }{6} ;\frac{11\pi }{6}](https://tex.z-dn.net/?f=%5Cfrac%7B7%5Cpi+%7D%7B6%7D+%3B%5Cfrac%7B11%5Cpi+%7D%7B6%7D)
![1-\frac{6}{x}>\frac{2}{1-x} \\\\1-\frac{6}{x}>\frac{2}{1-x}\\\\\frac{x-6}{x}-\frac{2}{1-x}>0\\\\\frac{x-x^{2}-6+6x-2x }{x(1-x)}>0\\\\\frac{-x^{2} +5x-6}{x(1-x)} >0\\\\\frac{x^{2}-5x+6 }{x(x-1)}>0\\\\x(x-2)(x-3)(x-1)>0](https://tex.z-dn.net/?f=1-%5Cfrac%7B6%7D%7Bx%7D%3E%5Cfrac%7B2%7D%7B1-x%7D+%5C%5C%5C%5C1-%5Cfrac%7B6%7D%7Bx%7D%3E%5Cfrac%7B2%7D%7B1-x%7D%5C%5C%5C%5C%5Cfrac%7Bx-6%7D%7Bx%7D-%5Cfrac%7B2%7D%7B1-x%7D%3E0%5C%5C%5C%5C%5Cfrac%7Bx-x%5E%7B2%7D-6%2B6x-2x+%7D%7Bx%281-x%29%7D%3E0%5C%5C%5C%5C%5Cfrac%7B-x%5E%7B2%7D+%2B5x-6%7D%7Bx%281-x%29%7D+%3E0%5C%5C%5C%5C%5Cfrac%7Bx%5E%7B2%7D-5x%2B6+%7D%7Bx%28x-1%29%7D%3E0%5C%5C%5C%5Cx%28x-2%29%28x-3%29%28x-1%29%3E0)
+ - + - +
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0 1 2 3
x ∈ (- ∞ ; 0) ∪ (1 ; 2) ∪ (3 ; + ∞)