16^(sin(x)*cos(x))=4

$$16^{sin{left (x right )} cos{left (x right )}} = 4$$

Дано уравнение
$$16^{sin{left (x right )} cos{left (x right )}} = 4$$
преобразуем
$$4^{sin{left (2 x right )}} – 4 = 0$$
$$16^{frac{1}{2} sin{left (2 x right )}} – 4 = 0$$
Сделаем замену
$$w = sin{left (2 x right )}$$
$$16^{frac{w}{2}} – 4 = 0$$
или
$$16^{frac{w}{2}} – 4 = 0$$
или
$$4^{w} = 4$$
или
$$4^{w} = 4$$
– это простейшее показательное ур-ние
Сделаем замену
$$v = 4^{w}$$
получим
$$v – 4 = 0$$
или
$$v – 4 = 0$$
Переносим свободные слагаемые (без v)
из левой части в правую, получим:
$$v = 4$$
Получим ответ: v = 4
делаем обратную замену
$$4^{w} = v$$
или
$$w = frac{log{left (v right )}}{log{left (4 right )}}$$
Тогда, окончательный ответ
$$w_{1} = frac{log{left (4 right )}}{log{left (4 right )}} = 1$$
делаем обратную замену
$$sin{left (2 x right )} = w$$
Дано уравнение
$$sin{left (2 x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$2 x = 2 pi n + {asin}{left (w right )}$$
$$2 x = 2 pi n – {asin}{left (w right )} + pi$$
Или
$$2 x = 2 pi n + {asin}{left (w right )}$$
$$2 x = 2 pi n – {asin}{left (w right )} + pi$$
, где n – любое целое число
Разделим обе части полученного ур-ния на
$$2$$
подставляем w:
$$x_{1} = pi n + frac{1}{2} {asin}{left (w_{1} right )}$$
$$x_{1} = pi n + frac{1}{2} {asin}{left (1 right )}$$
$$x_{1} = pi n + frac{pi}{4}$$
$$x_{2} = pi n – frac{1}{2} {asin}{left (w_{1} right )} + frac{pi}{2}$$
$$x_{2} = pi n – frac{1}{2} {asin}{left (1 right )} + frac{pi}{2}$$
$$x_{2} = pi n + frac{pi}{4}$$

/ __________________________________________
| / ________________ |
| / / 2 |
| / / 4*log (2) ________|
| ________________ 2* / – / -1 + ——— *log(4) + log(4) */ log(2) |
| / 2 / / 2 |
| / 4*log (2) / / log (4) |
x1 = 2*atan|-1 + / -1 + ——— + ————————————————————–|
| / 2 log(4) |
/ log (4) /

$$x_{1} = 2 {atan}{left (-1 + sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} + frac{2 sqrt{log{left (2 right )}}}{log{left (4 right )}} sqrt{- sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} log{left (4 right )} + log{left (4 right )}} right )}$$

/ ________________________________________
| / ________________ |
| / / 2 |
| / / 4*log (2) ________|
| ________________ 2* / / -1 + ——— *log(4) + log(4) */ log(2) |
| / 2 / / 2 |
| / 4*log (2) / / log (4) |
x2 = -2*atan|1 + / -1 + ——— – ————————————————————|
| / 2 log(4) |
/ log (4) /

$$x_{2} = – 2 {atan}{left (- frac{2 sqrt{log{left (2 right )}}}{log{left (4 right )}} sqrt{sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} log{left (4 right )} + log{left (4 right )}} + sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} + 1 right )}$$

/ __________________________________________
| / ________________ |
| / / 2 |
| / / 4*log (2) ________|
| ________________ 2* / – / -1 + ——— *log(4) + log(4) */ log(2) |
| / 2 / / 2 |
| / 4*log (2) / / log (4) |
x3 = -2*atan|1 – / -1 + ——— + ————————————————————–|
| / 2 log(4) |
/ log (4) /

$$x_{3} = – 2 {atan}{left (- sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} + 1 + frac{2 sqrt{log{left (2 right )}}}{log{left (4 right )}} sqrt{- sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} log{left (4 right )} + log{left (4 right )}} right )}$$

/ ________________________________________
| / ________________ |
| / / 2 |
| / / 4*log (2) ________|
| ________________ 2* / / -1 + ——— *log(4) + log(4) */ log(2) |
| / 2 / / 2 |
| / 4*log (2) / / log (4) |
x4 = -2*atan|1 + / -1 + ——— + ————————————————————|
| / 2 log(4) |
/ log (4) /

$$x_{4} = – 2 {atan}{left (sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} + 1 + frac{2 sqrt{log{left (2 right )}}}{log{left (4 right )}} sqrt{sqrt{-1 + frac{4 log^{2}{left (2 right )}}{log^{2}{left (4 right )}}} log{left (4 right )} + log{left (4 right )}} right )}$$

x1 = -90.3207886713000

x2 = -77.7544181730000

x3 = 32.2013247419000

x4 = -278.816347696000

x5 = -62.0464548656000

x6 = 0.785398174752000

x7 = 25.9181394572000

x8 = 82.4668070550000

x9 = 91.8915851960000

x10 = -40.0553062843000

x11 = -49.4800844057000

x12 = 76.1836219126000

x13 = -27.4889358294000

x14 = -96.6039742765000

x15 = -11.7809724256000

x16 = 66.7588438608000

x17 = -52.6216771815000

x18 = 63.6172513452000

x19 = 98.1747704997000

x20 = -87.1791960519000

x21 = -24.3473429484000

x22 = 47.9092880371000

x23 = -93.4623815579000

x24 = -21.2057503844000

x25 = -5.49778725287000

x26 = -99.7455667547000

x27 = 85.6083999179000

x28 = 38.4845099007000

x29 = 22.7765467354000

x30 = 10.2101761578000

x31 = 3.92699087708000

x32 = -74.6128257290000

x33 = -2.35619437464000

x34 = 19.6349541989000

x35 = -68.3296400968000

x36 = -43.1968989420000

x37 = -46.3384915225000

x38 = -65.1880474978000

x39 = -71.4712329819000

x40 = 69.9004366167000

x41 = 41.6261027722000

x42 = -84.0376034474000

x43 = -18.0641577034000

x44 = 16.4933613238000

x45 = -33.7721210085000

x46 = 57.3340656279000

x47 = -30.6305286326000

x48 = 13.3517684773000

x49 = 60.4756584778000

x50 = -55.7632695910000

x51 = -3712.57711834000

x52 = 79.3252142356000

x53 = 54.1924733268000

x54 = 35.3429170449000

x55 = 44.7676952975000

x56 = -8.63938008059000

x57 = 88.7499924253000