-4*cos(45)*cos(45)+x*cos(45)*sin(30)+y*cos(45)*cos(30)-z*cos(45)=0 4*cos(45)*sin(45)-x*cos(45)*cos(30)+y*cos(45)*cos(30)=0 -4*sin(45)-x*sin(45)-y*sin(45)-z*sin(30)=0

$$- z cos{left (45 right )} + y cos{left (45 right )} cos{left (30 right )} + x cos{left (45 right )} sin{left (30 right )} + – 4 cos{left (45 right )} cos{left (45 right )} = 0$$

4*cos(45)*sin(45) – x*cos(45)*cos(30) + y*cos(45)*cos(30) = 0

$$y cos{left (45 right )} cos{left (30 right )} + – x cos{left (30 right )} cos{left (45 right )} + sin{left (45 right )} 4 cos{left (45 right )} = 0$$

-4*sin(45) – x*sin(45) – y*sin(45) – z*sin(30) = 0

$$- z sin{left (30 right )} + – y sin{left (45 right )} + – x sin{left (45 right )} – 4 sin{left (45 right )} = 0$$

$$x_{1} = frac{1}{left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{4}{left (30 right )} sin{left (frac{pi}{4} + 30 right )}} left(2 sqrt{2} left(- 2 left(- cos{left (60 right )} + 1right) sin{left (45 right )} cos{left (frac{pi}{4} + 30 right )} + left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{2}{left (30 right )} cos{left (30 right )}right) sin{left (30 right )} sin{left (45 right )} tan{left (30 right )} + 4 left(- 2 sin{left (45 right )} + 2 sin{left (45 right )} cos{left (60 right )} + sin{left (45 right )} sin^{2}{left (60 right )} – sqrt{2} sin^{4}{left (30 right )} cos{left (frac{pi}{4} + 60 right )} + sin^{4}{left (30 right )} + 4 sin^{4}{left (30 right )} sin{left (45 right )}right) cos{left (45 right )} – 4 sqrt{2} left(- cos{left (60 right )} + 1right) sin^{2}{left (30 right )} sin{left (45 right )} sin{left (frac{pi}{4} + 30 right )}right)$$
=
$$frac{1}{left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{4}{left (30 right )} sin{left (frac{pi}{4} + 30 right )}} left(- 2 sqrt{2} left(- left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{2}{left (30 right )} cos{left (30 right )} + 2 left(- cos{left (60 right )} + 1right) sin{left (45 right )} cos{left (frac{pi}{4} + 30 right )}right) sin{left (30 right )} sin{left (45 right )} tan{left (30 right )} + 4 left(- 2 sin{left (45 right )} + 2 sin{left (45 right )} cos{left (60 right )} + sin{left (45 right )} sin^{2}{left (60 right )} – sqrt{2} sin^{4}{left (30 right )} cos{left (frac{pi}{4} + 60 right )} + sin^{4}{left (30 right )} + 4 sin^{4}{left (30 right )} sin{left (45 right )}right) cos{left (45 right )} – 4 sqrt{2} left(- cos{left (60 right )} + 1right) sin^{2}{left (30 right )} sin{left (45 right )} sin{left (frac{pi}{4} + 30 right )}right)$$
=

3.93285976192943

$$z_{1} = frac{8 sin{left (45 right )}}{left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{2}{left (30 right )}} left(- 2 sin{left (30 right )} sin{left (45 right )} cos{left (frac{pi}{4} + 30 right )} tan{left (30 right )} + sqrt{2} left(-1 + cos{left (60 right )}right) cos{left (45 right )} – 2 sin^{2}{left (30 right )} sin{left (frac{pi}{4} + 30 right )}right)$$
=
$$- frac{8 sin{left (45 right )}}{left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{2}{left (30 right )}} left(2 sin^{2}{left (30 right )} sin{left (frac{pi}{4} + 30 right )} + sqrt{2} left(- cos{left (60 right )} + 1right) cos{left (45 right )} + 2 sin{left (30 right )} sin{left (45 right )} cos{left (frac{pi}{4} + 30 right )} tan{left (30 right )}right)$$
=

-8.78404255083384

$$y_{1} = frac{2 tan{left (30 right )}}{left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{3}{left (30 right )} sin{left (frac{pi}{4} + 30 right )}} left(sqrt{2} left(frac{1}{2} cos{left (frac{pi}{4} + 120 right )} – cos{left (frac{pi}{4} + 60 right )} – frac{sqrt{2}}{2} cos{left (60 right )} + frac{3 sqrt{2}}{4}right) cos{left (30 right )} cos{left (45 right )} – 4 sqrt{2} sin^{2}{left (30 right )} sin{left (45 right )} sin{left (frac{pi}{4} + 30 right )} cos{left (30 right )} – 2 left(2 sqrt{2} sin{left (45 right )} sin{left (frac{pi}{4} + 30 right )} + sin{left (30 right )} – sqrt{2} sin{left (30 right )} cos{left (frac{pi}{4} + 60 right )}right) sin^{2}{left (30 right )} sin{left (45 right )}right)$$
=
$$- frac{tan{left (30 right )}}{2 left(- 2 cos{left (frac{pi}{4} + 60 right )} + sqrt{2} + 4 sqrt{2} sin{left (45 right )}right) sin^{3}{left (30 right )} sin{left (frac{pi}{4} + 30 right )}} left(8 left(2 sqrt{2} sin{left (45 right )} sin{left (frac{pi}{4} + 30 right )} + sin{left (30 right )} – sqrt{2} sin{left (30 right )} cos{left (frac{pi}{4} + 60 right )}right) sin^{2}{left (30 right )} sin{left (45 right )} + 16 sqrt{2} sin^{2}{left (30 right )} sin{left (45 right )} sin{left (frac{pi}{4} + 30 right )} cos{left (30 right )} – sqrt{2} left(2 cos{left (frac{pi}{4} + 120 right )} – 4 cos{left (frac{pi}{4} + 60 right )} – 2 sqrt{2} cos{left (60 right )} + 3 sqrt{2}right) cos{left (30 right )} cos{left (45 right )}right)$$
=

-18.1325023504979

$$- z cos{left (45 right )} + y cos{left (45 right )} cos{left (30 right )} + x cos{left (45 right )} sin{left (30 right )} + – 4 cos{left (45 right )} cos{left (45 right )} = 0$$
$$y cos{left (45 right )} cos{left (30 right )} + – x cos{left (30 right )} cos{left (45 right )} + sin{left (45 right )} 4 cos{left (45 right )} = 0$$
$$- z sin{left (30 right )} + – y sin{left (45 right )} + – x sin{left (45 right )} – 4 sin{left (45 right )} = 0$$

Приведём систему ур-ний к каноническому виду
$$x sin{left (30 right )} cos{left (45 right )} + y cos{left (30 right )} cos{left (45 right )} – z cos{left (45 right )} – 4 cos^{2}{left (45 right )} = 0$$
$$- x cos{left (30 right )} cos{left (45 right )} + y cos{left (30 right )} cos{left (45 right )} + 4 sin{left (45 right )} cos{left (45 right )} = 0$$
$$- x sin{left (45 right )} – y sin{left (45 right )} – z sin{left (30 right )} – 4 sin{left (45 right )} = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}x_{3} left(- cos{left (45 right )}right) + x_{1} sin{left (30 right )} cos{left (45 right )} + x_{2} cos{left (30 right )} cos{left (45 right )} x_{3} + x_{1} left(- cos{left (30 right )} cos{left (45 right )}right) + x_{2} cos{left (30 right )} cos{left (45 right )}x_{3} left(- sin{left (30 right )}right) + x_{1} left(- sin{left (45 right )}right) + x_{2} left(- sin{left (45 right )}right)end{matrix}right] = left[begin{matrix}4 cos^{2}{left (45 right )} – 4 sin{left (45 right )} cos{left (45 right )}4 sin{left (45 right )}end{matrix}right]$$
– это есть система уравнений, имеющая форму
A*x = B

Решение такого матричного ур-ния методом Крамера найдём так:

Т.к. определитель матрицы:
$$A = {det}{left (left[begin{matrix}sin{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 0 – sin{left (45 right )} & – sin{left (45 right )} & – sin{left (30 right )}end{matrix}right] right )} = – 2 sin{left (45 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin^{2}{left (30 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin{left (30 right )} cos^{2}{left (30 right )} cos^{2}{left (45 right )}$$
, 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{{det}{left (left[begin{matrix}4 cos^{2}{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} – 4 sin{left (45 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 04 sin{left (45 right )} & – sin{left (45 right )} & – sin{left (30 right )}end{matrix}right] right )}}{- 2 sin{left (45 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin^{2}{left (30 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin{left (30 right )} cos^{2}{left (30 right )} cos^{2}{left (45 right )}} = frac{1}{sin{left (30 right )} cos{left (45 right )}} left(frac{left(- frac{left(- 4 sin{left (45 right )} cos{left (45 right )} + frac{4 cos^{2}{left (45 right )}}{sin{left (30 right )}} cos{left (30 right )}right) left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right)}{frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}} + frac{4 cos{left (45 right )}}{sin{left (30 right )}} sin{left (45 right )} + 4 sin{left (45 right )}right) cos{left (45 right )}}{- frac{sin{left (45 right )}}{sin{left (30 right )}} – sin{left (30 right )} + frac{left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right) cos{left (30 right )} cos{left (45 right )}}{left(frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}right) sin{left (30 right )}}} + 4 cos^{2}{left (45 right )} – frac{cos{left (30 right )} cos{left (45 right )}}{frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}} left(- 4 sin{left (45 right )} cos{left (45 right )} + frac{4 cos^{2}{left (45 right )}}{sin{left (30 right )}} cos{left (30 right )} + frac{left(- frac{left(- 4 sin{left (45 right )} cos{left (45 right )} + frac{4 cos^{2}{left (45 right )}}{sin{left (30 right )}} cos{left (30 right )}right) left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right)}{frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}} + frac{4 cos{left (45 right )}}{sin{left (30 right )}} sin{left (45 right )} + 4 sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{left(- frac{sin{left (45 right )}}{sin{left (30 right )}} – sin{left (30 right )} + frac{left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right) cos{left (30 right )} cos{left (45 right )}}{left(frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}right) sin{left (30 right )}}right) sin{left (30 right )}}right)right)$$
=
$$frac{- 2 sqrt{2} cos{left (frac{pi}{4} + 105 right )} – 8 sin{left (45 right )} cos{left (30 right )} + 2 sqrt{2} sin{left (frac{pi}{4} + 15 right )} + 8 sin^{2}{left (45 right )}}{left(sin{left (60 right )} – cos{left (60 right )} + 1 + 4 sin{left (45 right )}right) cos{left (30 right )}}$$
$$x_{2} = frac{{det}{left (left[begin{matrix}sin{left (30 right )} cos{left (45 right )} & 4 cos^{2}{left (45 right )} & – cos{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & – 4 sin{left (45 right )} cos{left (45 right )} & 0 – sin{left (45 right )} & 4 sin{left (45 right )} & – sin{left (30 right )}end{matrix}right] right )}}{- 2 sin{left (45 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin^{2}{left (30 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin{left (30 right )} cos^{2}{left (30 right )} cos^{2}{left (45 right )}} = frac{1}{frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}} left(- 4 sin{left (45 right )} cos{left (45 right )} + frac{4 cos^{2}{left (45 right )}}{sin{left (30 right )}} cos{left (30 right )} + frac{left(- frac{left(- 4 sin{left (45 right )} cos{left (45 right )} + frac{4 cos^{2}{left (45 right )}}{sin{left (30 right )}} cos{left (30 right )}right) left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right)}{frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}} + frac{4 cos{left (45 right )}}{sin{left (30 right )}} sin{left (45 right )} + 4 sin{left (45 right )}right) cos{left (30
right )} cos{left (45 right )}}{left(- frac{sin{left (45 right )}}{sin{left (30 right )}} – sin{left (30 right )} + frac{left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right) cos{left (30 right )} cos{left (45 right )}}{left(frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}right) sin{left (30 right )}}right) sin{left (30 right )}}right)$$
=
$$frac{4 sin{left (75 right )} – 4 cos{left (90 right )} + 4 sin{left (15 right )} + 4 sin{left (45 right )} – 4 sin{left (105 right )} + 4}{left(- 4 sin{left (45 right )} – 1 + sqrt{2} cos{left (frac{pi}{4} + 60 right )}right) cos{left (30 right )}}$$
$$x_{3} = frac{{det}{left (left[begin{matrix}sin{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 4 cos^{2}{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & – 4 sin{left (45 right )} cos{left (45 right )} – sin{left (45 right )} & – sin{left (45 right )} & 4 sin{left (45 right )}end{matrix}right] right )}}{- 2 sin{left (45 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin^{2}{left (30 right )} cos{left (30 right )} cos^{2}{left (45 right )} – sin{left (30 right )} cos^{2}{left (30 right )} cos^{2}{left (45 right )}} = frac{- frac{left(- 4 sin{left (45 right )} cos{left (45 right )} + frac{4 cos^{2}{left (45 right )}}{sin{left (30 right )}} cos{left (30 right )}right) left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right)}{frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}} + frac{4 cos{left (45 right )}}{sin{left (30 right )}} sin{left (45 right )} + 4 sin{left (45 right )}}{- frac{sin{left (45 right )}}{sin{left (30 right )}} – sin{left (30 right )} + frac{left(- sin{left (45 right )} + frac{sin{left (45 right )} cos{left (30 right )}}{sin{left (30 right )}}right) cos{left (30 right )} cos{left (45 right )}}{left(frac{cos^{2}{left (30 right )} cos{left (45 right )}}{sin{left (30 right )}} + cos{left (30 right )} cos{left (45 right )}right) sin{left (30 right )}}}$$
=
$$frac{8 sin{left (45 right )}}{- 4 sin{left (45 right )} – 1 + sqrt{2} cos{left (frac{pi}{4} + 60 right )}} left(frac{sin{left (60 right )}}{2 cos{left (30 right )}} + cos{left (30 right )} + 2 cos{left (45 right )} + frac{sqrt{2} cos{left (frac{pi}{4} + 30 right )}}{cos{left (30 right )}} sin{left (45 right )}right)$$

Дана система ур-ний
$$- z cos{left (45 right )} + y cos{left (45 right )} cos{left (30 right )} + x cos{left (45 right )} sin{left (30 right )} + – 4 cos{left (45 right )} cos{left (45 right )} = 0$$
$$y cos{left (45 right )} cos{left (30 right )} + – x cos{left (30 right )} cos{left (45 right )} + sin{left (45 right )} 4 cos{left (45 right )} = 0$$
$$- z sin{left (30 right )} + – y sin{left (45 right )} + – x sin{left (45 right )} – 4 sin{left (45 right )} = 0$$

Приведём систему ур-ний к каноническому виду
$$x sin{left (30 right )} cos{left (45 right )} + y cos{left (30 right )} cos{left (45 right )} – z cos{left (45 right )} – 4 cos^{2}{left (45 right )} = 0$$
$$- x cos{left (30 right )} cos{left (45 right )} + y cos{left (30 right )} cos{left (45 right )} + 4 sin{left (45 right )} cos{left (45 right )} = 0$$
$$- x sin{left (45 right )} – y sin{left (45 right )} – z sin{left (30 right )} – 4 sin{left (45 right )} = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}sin{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 0 & – 4 sin{left (45 right )} cos{left (45 right )} – sin{left (45 right )} & – sin{left (45 right )} & – sin{left (30 right )} & 4 sin{left (45 right )}end{matrix}right]$$
В 1 ом столбце
$$left[begin{matrix}sin{left (30 right )} cos{left (45 right )} – cos{left (30 right )} cos{left (45 right )} – sin{left (45 right )}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
2 го элемента равнялись нулю.
– Для этого берём 2 ую строку
$$left[begin{matrix}- cos{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 0 & – 4 sin{left (45 right )} cos{left (45 right )}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 1 ой строки вычитаем:
$$left[begin{matrix}sin{left (30 right )} cos{left (45 right )} – sin{left (30 right )} cos{left (45 right )} & – -1 sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} – 0 & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}end{matrix}right] = left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 0 & – 4 sin{left (45 right )} cos{left (45 right )} – sin{left (45 right )} & – sin{left (45 right )} & – sin{left (30 right )} & 4 sin{left (45 right )}end{matrix}right]$$
Из 3 ой строки вычитаем:
$$left[begin{matrix}- sin{left (45 right )} – – sin{left (45 right )} & – sin{left (45 right )} – sin{left (45 right )} & – 0 – sin{left (30 right )} & 4 sin{left (45 right )} – – frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right] = left[begin{matrix}0 & – 2 sin{left (45 right )} & – sin{left (30 right )} & 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & cos{left (30 right )} cos{left (45 right )} & 0 & – 4 sin{left (45 right )} cos{left (45 right )} & – 2 sin{left (45 right )} & – sin{left (30 right )} & 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
Во 2 ом столбце
$$left[begin{matrix}sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}\cos{left (30 right )} cos{left (45 right )} – 2 sin{left (45 right )}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
1 го элемента равнялись нулю.
– Для этого берём 1 ую строку
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 2 ой строки вычитаем:
$$left[begin{matrix}- cos{left (30 right )} cos{left (45 right )} – 0 & – cos{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – frac{-1 cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}end{matrix}right] = left[begin{matrix}- cos{left (30 right )} cos{left (45 right )} & 0 & frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & 0 & frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}
& – 2 sin{left (45 right )} & – sin{left (30 right )} & 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
Из 3 ой строки вычитаем:
$$left[begin{matrix}- 0 & – 2 sin{left (45 right )} – – 2 sin{left (45 right )} & – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – frac{-1 cdot 2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right] = left[begin{matrix}0 & 0 & – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & – cos{left (45 right )} & 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & 0 & frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & 0 & – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
В 3 ом столбце
$$left[begin{matrix}- cos{left (45 right )}\frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
3 го элемента равнялись нулю.
– Для этого берём 3 ую строку
$$left[begin{matrix}0 & 0 & – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 1 ой строки вычитаем:
$$left[begin{matrix}- 0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} – 0 & – cos{left (45 right )} – – cos{left (45 right )} & – frac{1}{- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}} left(-1 left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (45 right )}right) + 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}end{matrix}right] = left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & 0 & frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (45 right )}}{- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}} + 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & 0 & frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (45 right )}}{- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}} + 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & 0 & frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & 0 & – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left
(45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$
Из 2 ой строки вычитаем:
$$left[begin{matrix}- cos{left (30 right )} cos{left (45 right )} – 0 & – 0 & frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} – frac{cos{left (30 right )} cos^{2}{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & – frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (30 right )} cos^{2}{left (45 right )}}{left(sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}right) left(- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}right)} + – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}end{matrix}right] = left[begin{matrix}- cos{left (30 right )} cos{left (45 right )} & 0 & 0 & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (30 right )} cos^{2}{left (45 right )}}{left(sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}right) left(- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}right)} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}end{matrix}right]$$
получаем
$$left[begin{matrix}0 & sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )} & 0 & frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (45 right )}}{- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}} + 4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – cos{left (30 right )} cos{left (45 right )} & 0 & 0 & – 4 sin{left (45 right )} cos{left (45 right )} – frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (30 right )} cos^{2}{left (45 right )}}{left(sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}right) left(- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}right)} – frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & 0 & – sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} & frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}end{matrix}right]$$

Все почти готово – осталось только найти неизвестные, решая элементарные ур-ния:
$$x_{2} left(sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}right) + frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )} – 4 cos^{2}{left (45 right )} – frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (45 right )}}{- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}} = 0$$
$$- x_{1} cos{left (30 right )} cos{left (45 right )} + frac{left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) cos{left (30 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + frac{left(frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} + 4 sin{left (45 right )} + frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}}right) cos{left (30 right )} cos^{2}{left (45 right )}}{left(sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}right) left(- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos
{left (45 right )}}right)} + 4 sin{left (45 right )} cos{left (45 right )} = 0$$
$$x_{3} left(- sin{left (30 right )} – frac{2 sin{left (45 right )} cos{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}}right) – frac{4 sin^{2}{left (45 right )}}{cos{left (30 right )}} – 4 sin{left (45 right )} – frac{2 left(4 cos^{2}{left (45 right )} – frac{4 cos{left (45 right )}}{cos{left (30 right )}} sin{left (30 right )} sin{left (45 right )}right) sin{left (45 right )}}{sin{left (30 right )} cos{left (45 right )} + cos{left (30 right )} cos{left (45 right )}} = 0$$
Получаем ответ:
$$x_{2} = frac{4 sin{left (75 right )} – 4 cos{left (90 right )} + 4 sin{left (15 right )} + 4 sin{left (45 right )} – 4 sin{left (105 right )} + 4}{left(- 4 sin{left (45 right )} – 1 + sqrt{2} cos{left (frac{pi}{4} + 60 right )}right) cos{left (30 right )}}$$
$$x_{1} = frac{- 2 sqrt{2} cos{left (frac{pi}{4} + 105 right )} – 8 sin{left (45 right )} cos{left (30 right )} + 2 sqrt{2} sin{left (frac{pi}{4} + 15 right )} + 8 sin^{2}{left (45 right )}}{left(sin{left (60 right )} – cos{left (60 right )} + 1 + 4 sin{left (45 right )}right) cos{left (30 right )}}$$
$$x_{3} = frac{8 sin{left (45 right )}}{- 4 sin{left (45 right )} – 1 + sqrt{2} cos{left (frac{pi}{4} + 60 right )}} left(frac{sin{left (60 right )}}{2 cos{left (30 right )}} + cos{left (30 right )} + 2 cos{left (45 right )} + frac{sqrt{2} cos{left (frac{pi}{4} + 30 right )}}{cos{left (30 right )}} sin{left (45 right )}right)$$

x1 = 3.932859761929435
y1 = -18.13250235049785
z1 = -8.784042550833844