sin(pi*(5*x+15)/6)=1/2 (уравнение)

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    Найду корень уравнения: sin(pi*(5*x+15)/6)=1/2

    Решение

    Вы ввели [src]
       /pi*(5*x + 15)\      
    sin|-------------| = 1/2
       \      6      /      
    sin(π(5x+15)6)=12\sin{\left(\frac{\pi \left(5 x + 15\right)}{6} \right)} = \frac{1}{2}
    Подробное решение
    Дано уравнение
    sin(π(5x+15)6)=12\sin{\left(\frac{\pi \left(5 x + 15\right)}{6} \right)} = \frac{1}{2}
    - это простейшее тригонометрическое ур-ние
    Это ур-ние преобразуется в
    5πx6=πn+acos(12)\frac{5 \pi x}{6} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}
    5πx6=πnπ+acos(12)\frac{5 \pi x}{6} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}
    Или
    5πx6=πn+π3\frac{5 \pi x}{6} = \pi n + \frac{\pi}{3}
    5πx6=πn2π3\frac{5 \pi x}{6} = \pi n - \frac{2 \pi}{3}
    , где n - любое целое число
    Разделим обе части полученного ур-ния на
    5π6\frac{5 \pi}{6}
    получим ответ:
    x1=6(πn+π3)5πx_{1} = \frac{6 \left(\pi n + \frac{\pi}{3}\right)}{5 \pi}
    x2=6(πn2π3)5πx_{2} = \frac{6 \left(\pi n - \frac{2 \pi}{3}\right)}{5 \pi}
    График
    0-80-60-40-2020406080-1001002-2
    Быстрый ответ [src]
    x1 = 2/5
    x1=25x_{1} = \frac{2}{5}
    x2 = 2
    x2=2x_{2} = 2
    Сумма и произведение корней [src]
    сумма
    0 + 2/5 + 2
    (0+25)+2\left(0 + \frac{2}{5}\right) + 2
    =
    12/5
    125\frac{12}{5}
    произведение
    1*2/5*2
    12521 \cdot \frac{2}{5} \cdot 2
    =
    4/5
    45\frac{4}{5}
    Численный ответ [src]
    x1 = 82.0
    x2 = 46.0
    x3 = 50.8
    x4 = 2.0
    x5 = 50.0
    x6 = 0.4
    x7 = 70.0
    x8 = 91.6
    x9 = -23.6
    x10 = -2.0
    x11 = 100.4
    x12 = -19.6
    x13 = -34.0
    x14 = 67.6
    x15 = 52.4
    x16 = 86.8
    x17 = 12.4
    x18 = -16.4
    x19 = 31.6
    x20 = 218.8
    x21 = 55.6
    x22 = 88.4
    x23 = -70.0
    x24 = -95.6
    x25 = 24.4
    x26 = 36.4
    x27 = 72.4
    x28 = 84.4
    x29 = -46.0
    x30 = 170.8
    x31 = -76.4
    x32 = 76.4
    x33 = 40.4
    x34 = 60.4
    x35 = 38.0
    x36 = -64.4
    x37 = -59.6
    x38 = -71.6
    x39 = -38.0
    x40 = -74.0
    x41 = 94.0
    x42 = -43.6
    x43 = -28.4
    x44 = -31.6
    x45 = 74.8
    x46 = 58.0
    x47 = 194.8
    x48 = 28.4
    x49 = 48.4
    x50 = -91.6
    x51 = 79.6
    x52 = -62.0
    x53 = 122.8
    x54 = 22.0
    x55 = 158.8
    x56 = 98.8
    x57 = -7.6
    x58 = 146.8
    x59 = -14.0
    x60 = -52.4
    x61 = -79.6
    x62 = 14.0
    x63 = 62.0
    x64 = 98.0
    x65 = -26.0
    x66 = 10.0
    x67 = -82.0
    x68 = -10.0
    x69 = 110.8
    x70 = -35.6
    x71 = -86.0
    x72 = 182.8
    x73 = -100.4
    x74 = -88.4
    x75 = -67.6
    x76 = 134.8
    x77 = 74.0
    x78 = -58.0
    x79 = -11.6
    x80 = 43.6
    x81 = 86.0
    x82 = 62.8
    x83 = -50.0
    x84 = 26.0
    x85 = -55.6
    x86 = -83.6
    x87 = 16.4
    x88 = -47.6
    x89 = 34.0
    x90 = -94.0
    x91 = 4.4
    x92 = -4.4
    x93 = -40.4
    x94 = -22.0
    x95 = 96.4
    x96 = 7.6
    x97 = 64.4
    x98 = 38.8
    x99 = 206.8
    x100 = 19.6
    x101 = -98.0
    График
    sin(pi*(5*x+15)/6)=1/2 (уравнение) /media/krcore-image-pods/hash/equation/b/24/ec8c7716d53cc2efc012cca33cacf.png