(-x)*cos(60)+y*cos(60)+4*cos(45)=0 x*sin(60)-y*sin(60)-z-4*sin(45)-8=0 x*cos(60)*3+7-16-z*4-16*cos(45)+8*sin(45)=0

$$- x cos{left (60 right )} + y cos{left (60 right )} + 4 cos{left (45 right )} = 0$$

x*sin(60) – y*sin(60) – z – 4*sin(45) – 8 = 0

$$- z + x sin{left (60 right )} – y sin{left (60 right )} – 4 sin{left (45 right )} – 8 = 0$$

x*cos(60)*3 + 7 – 16 – z*4 – 16*cos(45) + 8*sin(45) = 0

$$- 4 z + 3 x cos{left (60 right )} + 7 – 16 – 16 cos{left (45 right )} + 8 sin{left (45 right )} = 0$$

$$x_{1} = frac{1}{3 cos{left (60 right )}} left(-23 – 24 sin{left (45 right )} + frac{16 sqrt{2} cos{left (45 right )}}{cos{left (60 right )}} sin{left (frac{pi}{4} + 60 right )}right)$$
=
$$frac{1}{3 cos{left (60 right )}} left(-23 – 24 sin{left (45 right )} + frac{16 sqrt{2} cos{left (45 right )}}{cos{left (60 right )}} sin{left (frac{pi}{4} + 60 right )}right)$$
=

11.3139101667255

$$z_{1} = -8 + frac{4 sin{left (15 right )}}{cos{left (60 right )}}$$
=
$$-8 + frac{4 sin{left (15 right )}}{cos{left (60 right )}}$$
=

-10.7311170827330

$$y_{1} = frac{1}{3 cos^{2}{left (60 right )}} left(2 cos{left (15 right )} + 2 cos{left (105 right )} – 4 sin{left (105 right )} + 20 sin{left (15 right )} – 23 cos{left (60 right )}right)$$
=
$$frac{1}{3 cos^{2}{left (60 right )}} left(2 cos{left (15 right )} + 2 cos{left (105 right )} – 4 sin{left (105 right )} + 20 sin{left (15 right )} – 23 cos{left (60 right )}right)$$
=

13.5201883238701

$$- x cos{left (60 right )} + y cos{left (60 right )} + 4 cos{left (45 right )} = 0$$
$$- z + x sin{left (60 right )} – y sin{left (60 right )} – 4 sin{left (45 right )} – 8 = 0$$
$$- 4 z + 3 x cos{left (60 right )} + 7 – 16 – 16 cos{left (45 right )} + 8 sin{left (45 right )} = 0$$

Приведём систему ур-ний к каноническому виду
$$- x cos{left (60 right )} + y cos{left (60 right )} + 4 cos{left (45 right )} = 0$$
$$x sin{left (60 right )} – y sin{left (60 right )} – z – 8 – 4 sin{left (45 right )} = 0$$
$$3 x cos{left (60 right )} – 4 z – 9 – 16 cos{left (45 right )} + 8 sin{left (45 right )} = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}0 x_{3} + x_{1} left(- cos{left (60 right )}right) + x_{2} cos{left (60 right )} – x_{3} + x_{1} sin{left (60 right )} + x_{2} left(- sin{left (60 right )}right) – 4 x_{3} + x_{1} cdot 3 cos{left (60 right )} + 0 x_{2}end{matrix}right] = left[begin{matrix}- 4 cos{left (45 right )}4 sin{left (45 right )} + 8 – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right]$$
– это есть система уравнений, имеющая форму
A*x = B

Решение такого матричного ур-ния методом Крамера найдём так:

Т.к. определитель матрицы:
$$A = {det}{left (left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0\sin{left (60 right )} & – sin{left (60 right )} & -13 cos{left (60 right )} & 0 & -4end{matrix}right] right )} = – 3 cos^{2}{left (60 right )}$$
, то
Корень xi получается делением определителя матрицы Ai. на определитель матрицы A.
( Ai получаем заменой в матрице A i-го столбца на столбец B )
$$x_{1} = – frac{1}{3 cos^{2}{left (60 right )}} {det}{left (left[begin{matrix}- 4 cos{left (45 right )} & cos{left (60 right )} & 04 sin{left (45 right )} + 8 & – sin{left (60 right )} & -1 – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9 & 0 & -4end{matrix}right] right )} = – frac{1}{cos{left (60 right )}} left(- frac{16}{3} cos{left (45 right )} – frac{16 sin{left (60 right )} cos{left (45 right )}}{3 cos{left (60 right )}} + 8 sin{left (45 right )} + frac{23}{3}right)$$
=
$$frac{1}{3 cos{left (60 right )}} left(-23 – 24 sin{left (45 right )} + frac{16 sqrt{2} cos{left (45 right )}}{cos{left (60 right )}} sin{left (frac{pi}{4} + 60 right )}right)$$
$$x_{2} = – frac{1}{3 cos^{2}{left (60 right )}} {det}{left (left[begin{matrix}- cos{left (60 right )} & – 4 cos{left (45 right )} & 0\sin{left (60 right )} & 4 sin{left (45 right )} + 8 & -13 cos{left (60 right )} & – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9 & -4end{matrix}right] right )} = frac{1}{3 cos{left (60 right )}} left(-23 – 24 sin{left (45 right )} + 4 cos{left (45 right )} + frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )}right)$$
=
$$frac{1}{3 cos^{2}{left (60 right )}} left(16 sin{left (60 right )} cos{left (45 right )} – left(- 4 cos{left (45 right )} + 24 sin{left (45 right )} + 23right) cos{left (60 right )}right)$$
$$x_{3} = – frac{1}{3 cos^{2}{left (60 right )}} {det}{left (left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & – 4 cos{left (45 right )}\sin{left (60 right )} & – sin{left (60 right )} & 4 sin{left (45 right )} + 83 cos{left (60 right )} & 0 & – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right] right )} = -8 – 4 sin{left (45 right )} + frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )}$$
=
$$-8 + frac{4 sin{left (15 right )}}{cos{left (60 right )}}$$

Дана система ур-ний
$$- x cos{left (60 right )} + y cos{left (60 right )} + 4 cos{left (45 right )} = 0$$
$$- z + x sin{left (60 right )} – y sin{left (60 right )} – 4 sin{left (45 right )} – 8 = 0$$
$$- 4 z + 3 x cos{left (60 right )} + 7 – 16 – 16 cos{left (45 right )} + 8 sin{left (45 right )} = 0$$

Приведём систему ур-ний к каноническому виду
$$- x cos{left (60 right )} + y cos{left (60 right )} + 4 cos{left (45 right )} = 0$$
$$x sin{left (60 right )} – y sin{left (60 right )} – z – 8 – 4 sin{left (45 right )} = 0$$
$$3 x cos{left (60 right )} – 4 z – 9 – 16 cos{left (45 right )} + 8 sin{left (45 right )} = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )}\sin{left (60 right )} & – sin{left (60 right )} & -1 & 4 sin{left (45 right )} + 83 cos{left (60 right )} & 0 & -4 & – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right]$$
В 1 ом столбце
$$left[begin{matrix}- cos{left (60 right )}\sin{left (60 right )}3 cos{left (60 right )}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
1 го элемента равнялись нулю.
– Для этого берём 1 ую строку
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 2 ой строки вычитаем:
$$left[begin{matrix}sin{left (60 right )} – sin{left (60 right )} & – -1 sin{left (60 right )} – sin{left (60 right )} & -1 – 0 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 8end{matrix}right] = left[begin{matrix}0 & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 8end{matrix}right]$$
получаем
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )} & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 83 cos{left (60 right )} & 0 & -4 & – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right]$$
Из 3 ой строки вычитаем:
$$left[begin{matrix}3 cos{left (60 right )} – 3 cos{left (60 right )} & – -1 cdot 3 cos{left (60 right )} & -4 & – 12 cos{left (45 right )} + – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right] = left[begin{matrix}0 & 3 cos{left (60 right )} & -4 & – 8 sin{left (45 right )} + 4 cos{left (45 right )} + 9end{matrix}right]$$
получаем
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )} & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 8 & 3 cos{left (60 right )} & -4 & – 8 sin{left (45 right )} + 4 cos{left (45 right )} + 9end{matrix}right]$$
Во 2 ом столбце
$$left[begin{matrix}cos{left (60 right )}3 cos{left (60 right )}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
1 го элемента равнялись нулю.
– Для этого берём 1 ую строку
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 3 ой строки вычитаем:
$$left[begin{matrix}- -1 cdot 3 cos{left (60 right )} & 3 cos{left (60 right )} – 3 cos{left (60 right )} & -4 & – 8 sin{left (45 right )} + 4 cos{left (45 right )} + 9 – – 12 cos{left (45 right )}end{matrix}right] = left[begin{matrix}3 cos{left (60 right )} & 0 & -4 & – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right]$$
получаем
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )} & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 83 cos{left (60 right )} & 0 & -4 & – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right]$$
В 3 ом столбце
$$left[begin{matrix}0 -1 -4end{matrix}right]$$
делаем так, чтобы все элементы, кроме
2 го элемента равнялись нулю.
– Для этого берём 2 ую строку
$$left[begin{matrix}0 & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 8end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 3 ой строки вычитаем:
$$left[begin{matrix}3 cos{left (60 right )} & 0 & 0 & – – frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 16 sin{left (45 right )} + 32 + – 8 sin{left (45 right )} + 16 cos{left (45 right )} + 9end{matrix}right] = left[begin{matrix}3 cos{left (60 right )} & 0 & 0 & -23 – 24 sin{left (45 right )} + frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 16 cos{left (45 right )}end{matrix}right]$$
получаем
$$left[begin{matrix}- cos{left (60 right )} & cos{left (60 right )} & 0 & – 4 cos{left (45 right )} & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 83 cos{left (60 right )} & 0 & 0 & -23 – 24 sin{left (45 right )} + frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 16 cos{left (45 right )}end{matrix}right]$$
В 1 ом столбце
$$left[begin{matrix}- cos{left (60 right )}3 cos{left (60 right )}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
3 го элемента равнялись нулю.
– Для этого берём 3 ую строку
$$left[begin{matrix}3 cos{left (60 right )} & 0 & 0 & -23 – 24 sin{left (45 right )} + frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 16 cos{left (45 right )}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 1 ой строки вычитаем:
$$left[begin{matrix}- -1 cos{left (60 right )} – cos{left (60 right )} & cos{left (60 right )} & 0 & – – frac{16}{3} cos{left (45 right )} – frac{16 sin{left (60 right )} cos{left (45 right )}}{3 cos{left (60 right )}} + 8 sin{left (45 right )} + frac{23}{3} – 4 cos{left (45 right )}end{matrix}right] = left[begin{matrix}0 & cos{left (60 right )} & 0 & – frac{23}{3} – 8 sin{left (45 right )} + frac{4}{3} cos{left (45 right )} + frac{16 sin{left (60 right )} cos{left (45 right )}}{3 cos{left (60 right )}}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & cos{left (60 right )} & 0 & – frac{23}{3} – 8 sin{left (45 right )} + frac{4}{3} cos{left (45 right )} + frac{16 sin{left (60 right )} cos{left (45 right )}}{3 cos{left (60 right )}} & 0 & -1 & – frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 4 sin{left (45 right )} + 83 cos{left (60 right )} & 0 & 0 & -23 – 24 sin{left (45 right )} + frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 16 cos{left (45 right )}end{matrix}right]$$

Все почти готово
– осталось только найти неизвестные, решая элементарные ур-ния:
$$x_{2} cos{left (60 right )} – frac{16 sin{left (60 right )} cos{left (45 right )}}{3 cos{left (60 right )}} – frac{4}{3} cos{left (45 right )} + 8 sin{left (45 right )} + frac{23}{3} = 0$$
$$- x_{3} – 8 – 4 sin{left (45 right )} + frac{4 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} = 0$$
$$3 x_{1} cos{left (60 right )} – 16 cos{left (45 right )} – frac{16 cos{left (45 right )}}{cos{left (60 right )}} sin{left (60 right )} + 24 sin{left (45 right )} + 23 = 0$$
Получаем ответ:
$$x_{2} = frac{1}{3 cos^{2}{left (60 right )}} left(16 sin{left (60 right )} cos{left (45 right )} + left(-23 – 24 sin{left (45 right )} + 4 cos{left (45 right )}right) cos{left (60 right )}right)$$
$$x_{3} = -8 + frac{4 sin{left (15 right )}}{cos{left (60 right )}}$$
$$x_{1} = frac{1}{3 cos{left (60 right )}} left(-23 – 24 sin{left (45 right )} + frac{16 sqrt{2} cos{left (45 right )}}{cos{left (60 right )}} sin{left (frac{pi}{4} + 60 right )}right)$$

x1 = 11.31391016672553
y1 = 13.5201883238701
z1 = -10.73111708273298