-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

Дано

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

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

$$- z \sin{\left (30 \right )} + — y \sin{\left (45 \right )} + — x \sin{\left (45 \right )} — 4 \sin{\left (45 \right )} = 0$$
Ответ
$$x_{1} = \frac{1}{\left(- 2 \cos{\left (\frac{\pi}{4} + 60 \right )} + \sqrt{2} + 4 \sqrt{2} \sin{\left (45 \right )}\right) \sin^{4}{\left (30 \right )} \sin{\left (\frac{\pi}{4} + 30 \right )}} \left(2 \sqrt{2} \left(- 2 \left(- \cos{\left (60 \right )} + 1\right) \sin{\left (45 \right )} \cos{\left (\frac{\pi}{4} + 30 \right )} + \left(- 2 \cos{\left (\frac{\pi}{4} + 60 \right )} + \sqrt{2} + 4 \sqrt{2} \sin{\left (45 \right )}\right) \sin^{2}{\left (30 \right )} \cos{\left (30 \right )}\right) \sin{\left (30 \right )} \sin{\left (45 \right )} \tan{\left (30 \right )} + 4 \left(- 2 \sin{\left (45 \right )} + 2 \sin{\left (45 \right )} \cos{\left (60 \right )} + \sin{\left (45 \right )} \sin^{2}{\left (60 \right )} — \sqrt{2} \sin^{4}{\left (30 \right )} \cos{\left (\frac{\pi}{4} + 60 \right )} + \sin^{4}{\left (30 \right )} + 4 \sin^{4}{\left (30 \right )} \sin{\left (45 \right )}\right) \cos{\left (45 \right )} — 4 \sqrt{2} \left(- \cos{\left (60 \right )} + 1\right) \sin^{2}{\left (30 \right )} \sin{\left (45 \right )} \sin{\left (\frac{\pi}{4} + 30 \right )}\right)$$
=
$$\frac{1}{\left(- 2 \cos{\left (\frac{\pi}{4} + 60 \right )} + \sqrt{2} + 4 \sqrt{2} \sin{\left (45 \right )}\right) \sin^{4}{\left (30 \right )} \sin{\left (\frac{\pi}{4} + 30 \right )}} \left(- 2 \sqrt{2} \left(- \left(- 2 \cos{\left (\frac{\pi}{4} + 60 \right )} + \sqrt{2} + 4 \sqrt{2} \sin{\left (45 \right )}\right) \sin^{2}{\left (30 \right )} \cos{\left (30 \right )} + 2 \left(- \cos{\left (60 \right )} + 1\right) \sin{\left (45 \right )} \cos{\left (\frac{\pi}{4} + 30 \right )}\right) \sin{\left (30 \right )} \sin{\left (45 \right )} \tan{\left (30 \right )} + 4 \left(- 2 \sin{\left (45 \right )} + 2 \sin{\left (45 \right )} \cos{\left (60 \right )} + \sin{\left (45 \right )} \sin^{2}{\left (60 \right )} — \sqrt{2} \sin^{4}{\left (30 \right )} \cos{\left (\frac{\pi}{4} + 60 \right )} + \sin^{4}{\left (30 \right )} + 4 \sin^{4}{\left (30 \right )} \sin{\left (45 \right )}\right) \cos{\left (45 \right )} — 4 \sqrt{2} \left(- \cos{\left (60 \right )} + 1\right) \sin^{2}{\left (30 \right )} \sin{\left (45 \right )} \sin{\left (\frac{\pi}{4} + 30 \right )}\right)$$
=

3.93285976192943

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

-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

Метод Крамера
$$- z \cos{\left (45 \right )} + y \cos{\left (45 \right )} \cos{\left (30 \right )} + x \cos{\left (45 \right )} \sin{\left (30 \right )} + — 4 \cos{\left (45 \right )} \cos{\left (45 \right )} = 0$$
$$y \cos{\left (45 \right )} \cos{\left (30 \right )} + — x \cos{\left (30 \right )} \cos{\left (45 \right )} + \sin{\left (45 \right )} 4 \cos{\left (45 \right )} = 0$$
$$- z \sin{\left (30 \right )} + — y \sin{\left (45 \right )} + — x \sin{\left (45 \right )} — 4 \sin{\left (45 \right )} = 0$$

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
$$\left[begin{matrix}x_{3} \left(- \cos{\left (45 \right )}\right) + x_{1} \sin{\left (30 \right )} \cos{\left (45 \right )} + x_{2} \cos{\left (30 \right )} \cos{\left (45 \right )}\0 x_{3} + x_{1} \left(- \cos{\left (30 \right )} \cos{\left (45 \right )}\right) + x_{2} \cos{\left (30 \right )} \cos{\left (45 \right )}\x_{3} \left(- \sin{\left (30 \right )}\right) + x_{1} \left(- \sin{\left (45 \right )}\right) + x_{2} \left(- \sin{\left (45 \right )}\right)end{matrix}\right] = \left[begin{matrix}4 \cos^{2}{\left (45 \right )}\ — 4 \sin{\left (45 \right )} \cos{\left (45 \right )}\4 \sin{\left (45 \right )}end{matrix}\right]$$
— 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 )} & 0\4 \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)$$
=
$$\frac{4 \sin{\left (75 \right )} — 4 \cos{\left (90 \right )} + 4 \sin{\left (15 \right )} + 4 \sin{\left (45 \right )} — 4 \sin{\left (105 \right )} + 4}{\left(- 4 \sin{\left (45 \right )} — 1 + \sqrt{2} \cos{\left (\frac{\pi}{4} + 60 \right )}\right) \cos{\left (30 \right )}}$$
$$x_{3} = \frac{{det}{\left (\left[begin{matrix}\sin{\left (30 \right )} \cos{\left (45 \right )} & \cos{\left (30 \right )} \cos{\left (45 \right )} & 4 \cos^{2}{\left (45 \right )}\ — \cos{\left (30 \right )} \cos{\left (45 \right )} & \cos{\left (30 \right )} \cos{\left (45 \right )} & — 4 \sin{\left (45 \right )} \cos{\left (45 \right )}\ — \sin{\left (45 \right )} & — \sin{\left (45 \right )} & 4 \sin{\left (45 \right )}end{matrix}\right] \right )}}{- 2 \sin{\left (45 \right )} \cos{\left (30 \right )} \cos^{2}{\left (45 \right )} — \sin^{2}{\left (30 \right )} \cos{\left (30 \right )} \cos^{2}{\left (45 \right )} — \sin{\left (30 \right )} \cos^{2}{\left (30 \right )} \cos^{2}{\left (45 \right )}} = \frac{- \frac{\left(- 4 \sin{\left (45 \right )} \cos{\left (45 \right )} + \frac{4 \cos^{2}{\left (45 \right )}}{\sin{\left (30 \right )}} \cos{\left (30 \right )}\right) \left(- \sin{\left (45 \right )} + \frac{\sin{\left (45 \right )} \cos{\left (30 \right )}}{\sin{\left (30 \right )}}\right)}{\frac{\cos^{2}{\left (30 \right )} \cos{\left (45 \right )}}{\sin{\left (30 \right )}} + \cos{\left (30 \right )} \cos{\left (45 \right )}} + \frac{4 \cos{\left (45 \right )}}{\sin{\left (30 \right )}} \sin{\left (45 \right )} + 4 \sin{\left (45 \right )}}{- \frac{\sin{\left (45 \right )}}{\sin{\left (30 \right )}} — \sin{\left (30 \right )} + \frac{\left(- \sin{\left (45 \right )} + \frac{\sin{\left (45 \right )} \cos{\left (30 \right )}}{\sin{\left (30 \right )}}\right) \cos{\left (30 \right )} \cos{\left (45 \right )}}{\left(\frac{\cos^{2}{\left (30 \right )} \cos{\left (45 \right )}}{\sin{\left (30 \right )}} + \cos{\left (30 \right )} \cos{\left (45 \right )}\right) \sin{\left (30 \right )}}}$$
=
$$\frac{8 \sin{\left (45 \right )}}{- 4 \sin{\left (45 \right )} — 1 + \sqrt{2} \cos{\left (\frac{\pi}{4} + 60 \right )}} \left(\frac{\sin{\left (60 \right )}}{2 \cos{\left (30 \right )}} + \cos{\left (30 \right )} + 2 \cos{\left (45 \right )} + \frac{\sqrt{2} \cos{\left (\frac{\pi}{4} + 30 \right )}}{\cos{\left (30 \right )}} \sin{\left (45 \right )}\right)$$

Метод Гаусса
Читайте также  y=x^2 y=6-x
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
$$\left[begin{matrix}0 & \sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )} & — \cos{\left (45 \right )} & 4 \cos^{2}{\left (45 \right )} — \frac{4 \cos{\left (45 \right )}}{\cos{\left (30 \right )}} \sin{\left (30 \right )} \sin{\left (45 \right )}\ — \cos{\left (30 \right )} \cos{\left (45 \right )} & \cos{\left (30 \right )} \cos{\left (45 \right )} & 0 & — 4 \sin{\left (45 \right )} \cos{\left (45 \right )}\0 & — 2 \sin{\left (45 \right )} & — \sin{\left (30 \right )} & 4 \sin{\left (45 \right )} + \frac{4 \sin^{2}{\left (45 \right )}}{\cos{\left (30 \right )}}end{matrix}\right]$$
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
$$\left[begin{matrix}0 & \sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )} & — \cos{\left (45 \right )} & 4 \cos^{2}{\left (45 \right )} — \frac{4 \cos{\left (45 \right )}}{\cos{\left (30 \right )}} \sin{\left (30 \right )} \sin{\left (45 \right )}\ — \cos{\left (30 \right )} \cos{\left (45 \right )} & 0 & \frac{\cos{\left (30 \right )} \cos^{2}{\left (45 \right )}}{\sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )}} & — 4 \sin{\left (45 \right )} \cos{\left (45 \right )} — \frac{\left(4 \cos^{2}{\left (45 \right )} — \frac{4 \cos{\left (45 \right )}}{\cos{\left (30 \right )}} \sin{\left (30 \right )} \sin{\left (45 \right )}\right) \cos{\left (30 \right )} \cos{\left (45 \right )}}{\sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )}}\0 & — 2 \sin{\left (45 \right )} & — \sin{\left (30 \right )} & 4 \sin{\left (45 \right )} + \frac{4 \sin^{2}{\left (45 \right )}}{\cos{\left (30 \right )}}end{matrix}\right]$$
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
$$\left[begin{matrix}0 & \sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )} & — \cos{\left (45 \right )} & 4 \cos^{2}{\left (45 \right )} — \frac{4 \cos{\left (45 \right )}}{\cos{\left (30 \right )}} \sin{\left (30 \right )} \sin{\left (45 \right )}\ — \cos{\left (30 \right )} \cos{\left (45 \right )} & 0 & \frac{\cos{\left (30 \right )} \cos^{2}{\left (45 \right )}}{\sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )}} & — 4 \sin{\left (45 \right )} \cos{\left (45 \right )} — \frac{\left(4 \cos^{2}{\left (45 \right )} — \frac{4 \cos{\left (45 \right )}}{\cos{\left (30 \right )}} \sin{\left (30 \right )} \sin{\left (45 \right )}\right) \cos{\left (30 \right )} \cos{\left (45 \right )}}{\sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )}}\0 & 0 & — \sin{\left (30 \right )} — \frac{2 \sin{\left (45 \right )} \cos{\left (45 \right )}}{\sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )}} & \frac{2 \left(4 \cos^{2}{\left (45 \right )} — \frac{4 \cos{\left (45 \right )}}{\cos{\left (30 \right )}} \sin{\left (30 \right )} \sin{\left (45 \right )}\right) \sin{\left (45 \right )}}{\sin{\left (30 \right )} \cos{\left (45 \right )} + \cos{\left (30 \right )} \cos{\left (45 \right )}} + 4 \sin{\left (45 \right )} + \frac{4 \sin^{2}{\left (45 \right )}}{\cos{\left (30 \right )}}end{matrix}\right]$$
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 & 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 & 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]$$

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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:
$$x_{2} = \frac{4 \sin{\left (75 \right )} — 4 \cos{\left (90 \right )} + 4 \sin{\left (15 \right )} + 4 \sin{\left (45 \right )} — 4 \sin{\left (105 \right )} + 4}{\left(- 4 \sin{\left (45 \right )} — 1 + \sqrt{2} \cos{\left (\frac{\pi}{4} + 60 \right )}\right) \cos{\left (30 \right )}}$$
$$x_{1} = \frac{- 2 \sqrt{2} \cos{\left (\frac{\pi}{4} + 105 \right )} — 8 \sin{\left (45 \right )} \cos{\left (30 \right )} + 2 \sqrt{2} \sin{\left (\frac{\pi}{4} + 15 \right )} + 8 \sin^{2}{\left (45 \right )}}{\left(\sin{\left (60 \right )} — \cos{\left (60 \right )} + 1 + 4 \sin{\left (45 \right )}\right) \cos{\left (30 \right )}}$$
$$x_{3} = \frac{8 \sin{\left (45 \right )}}{- 4 \sin{\left (45 \right )} — 1 + \sqrt{2} \cos{\left (\frac{\pi}{4} + 60 \right )}} \left(\frac{\sin{\left (60 \right )}}{2 \cos{\left (30 \right )}} + \cos{\left (30 \right )} + 2 \cos{\left (45 \right )} + \frac{\sqrt{2} \cos{\left (\frac{\pi}{4} + 30 \right )}}{\cos{\left (30 \right )}} \sin{\left (45 \right )}\right)$$

Численный ответ
Читайте также  x*45/2-y*25/2=-14 y*45/2-x*25/2=14

x1 = 3.932859761929435
y1 = -18.13250235049785
z1 = -8.784042550833844

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