Дано

$$x_{1} sin{left (35 right )} + x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) = 3623$$

(1 – cos(35))*x1 + (cos(32) – cos(35))*x2 = 370

$$x_{1} left(- cos{left (35 right )} + 1right) + x_{2} left(cos{left (32 right )} – cos{left (35 right )}right) = 370$$
Подробное решение
Дана система ур-ний
$$x_{1} sin{left (35 right )} + x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) = 3623$$
$$x_{1} left(- cos{left (35 right )} + 1right) + x_{2} left(cos{left (32 right )} – cos{left (35 right )}right) = 370$$

Из 1-го ур-ния выразим x1
$$x_{1} sin{left (35 right )} + x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) = 3623$$
Перенесем слагаемое с переменной x2 из левой части в правую со сменой знака
$$x_{1} sin{left (35 right )} = – x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) + 3623$$
$$x_{1} sin{left (35 right )} = – x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) + 3623$$
Разделим обе части ур-ния на множитель при x1
$$frac{x_{1} sin{left (35 right )}}{sin{left (35 right )}} = frac{1}{sin{left (35 right )}} left(- x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) + 3623right)$$
$$x_{1} = frac{1}{sin{left (35 right )}} left(x_{2} left(- sin{left (35 right )} + sin{left (32 right )}right) + 3623right)$$
Подставим найденное x1 в 2-е ур-ние
$$x_{1} left(- cos{left (35 right )} + 1right) + x_{2} left(cos{left (32 right )} – cos{left (35 right )}right) = 370$$
Получим:
$$x_{2} left(cos{left (32 right )} – cos{left (35 right )}right) + frac{1}{sin{left (35 right )}} left(x_{2} left(- sin{left (35 right )} + sin{left (32 right )}right) + 3623right) left(- cos{left (35 right )} + 1right) = 370$$
$$frac{x_{2} sin{left (32 right )}}{sin{left (35 right )}} – frac{x_{2} cos{left (35 right )}}{sin{left (35 right )}} sin{left (32 right )} – x_{2} + x_{2} cos{left (32 right )} + frac{3623}{sin{left (35 right )}} – frac{3623 cos{left (35 right )}}{sin{left (35 right )}} = 370$$
Перенесем свободное слагаемое 3623/sin(35) – 3623*cos(35)/sin(35) из левой части в правую со сменой знака
$$frac{x_{2} sin{left (32 right )}}{sin{left (35 right )}} – frac{x_{2} cos{left (35 right )}}{sin{left (35 right )}} sin{left (32 right )} – x_{2} + x_{2} cos{left (32 right )} = 370 + – frac{-1 cdot 3623 cos{left (35 right )}}{sin{left (35 right )}} – frac{3623}{sin{left (35 right )}}$$
$$frac{x_{2} sin{left (32 right )}}{sin{left (35 right )}} – frac{x_{2} cos{left (35 right )}}{sin{left (35 right )}} sin{left (32 right )} – x_{2} + x_{2} cos{left (32 right )} = 370 + frac{3623 cos{left (35 right )}}{sin{left (35 right )}} – frac{3623}{sin{left (35 right )}}$$
Разделим обе части ур-ния на множитель при x2
$$frac{frac{x_{2} sin{left (32 right )}}{sin{left (35 right )}} – frac{x_{2} cos{left (35 right )}}{sin{left (35 right )}} sin{left (32 right )} – x_{2} + x_{2} cos{left (32 right )}}{frac{x_{2} sin{left (32 right )}}{sin{left (35 right )}} – frac{x_{2} cos{left (35 right )}}{sin{left (35 right )}} sin{left (32 right )} – x_{2} + x_{2} cos{left (32 right )}} = frac{370 + frac{3623 cos{left (35 right )}}{sin{left (35 right )}} – frac{3623}{sin{left (35 right )}}}{frac{x_{2} sin{left (32 right )}}{sin{left (35 right )}} – frac{x_{2} cos{left (35 right )}}{sin{left (35 right )}} sin{left (32 right )} – x_{2} + x_{2} cos{left (32 right )}}$$
$$frac{-3623 + 3623 cos{left (35 right )} + 370 sin{left (35 right )}}{x_{2} left(sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}right)} = 1$$
Т.к.
$$x_{1} = frac{1}{sin{left (35 right )}} left(x_{2} left(- sin{left (35 right )} + sin{left (32 right )}right) + 3623right)$$
то
$$x_{1} = frac{1}{sin{left (35 right )}} left(- sin{left (35 right )} + sin{left (32 right )} + 3623right)$$
$$x_{1} = frac{1}{sin{left (35 right )}} left(- sin{left (35 right )} + sin{left (32 right )} + 3623right)$$

Ответ:
$$x_{1} = frac{1}{sin{left (35 right )}} left(- sin{left (35 right )} + sin{left (32 right )} + 3623right)$$
$$frac{-3623 + 3623 cos{left (35 right )} + 370 sin{left (35 right )}}{x_{2} left(sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}right)} = 1$$

Ответ
$$x_{11} = frac{1}{left(sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}right) sin{left (35 right )}} left(frac{3623}{2} sin{left (67 right )} – frac{3623}{2} sin{left (70 right )} – 185 + 185 cos{left (3 right )} – 185 cos{left (67 right )} + 185 cos{left (70 right )} + frac{3623}{2} sin{left (3 right )}right)$$
=
$$frac{3623 sin{left (67 right )} – 3623 sin{left (70 right )} – 370 + 370 cos{left (3 right )} – 370 cos{left (67 right )} + 370 cos{left (70 right )} + 3623 sin{left (3 right )}}{2 left(sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}right) sin{left (35 right )}}$$
=

5941.59821161297

$$x_{21} = frac{-3623 + 3623 cos{left (35 right )} + 370 sin{left (35 right )}}{sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}}$$
=
$$frac{-3623 + 3623 cos{left (35 right )} + 370 sin{left (35 right )}}{sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}}$$
=

-6295.4578564169

Метод Крамера
$$x_{1} sin{left (35 right )} + x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) = 3623$$
$$x_{1} left(- cos{left (35 right )} + 1right) + x_{2} left(cos{left (32 right )} – cos{left (35 right )}right) = 370$$

Приведём систему ур-ний к каноническому виду
$$x_{1} sin{left (35 right )} – x_{2} sin{left (32 right )} + x_{2} sin{left (35 right )} – 3623 = 0$$
$$- x_{1} cos{left (35 right )} + x_{1} + x_{2} cos{left (32 right )} – x_{2} cos{left (35 right )} – 370 = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}x_{1} sin{left (35 right )} + x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right)x_{1} left(- cos{left (35 right )} + 1right) + x_{2} left(cos{left (32 right )} – cos{left (35 right )}right)end{matrix}right] = left[begin{matrix}3623370end{matrix}right]$$
– это есть система уравнений, имеющая форму
A*x = B

Решение такого матричного ур-ния методом Крамера найдём так:

Т.к. определитель матрицы:
$$A = {det}{left (left[begin{matrix}sin{left (35 right )} & – sin{left (32 right )} + sin{left (35 right )} – cos{left (35 right )} + 1 & cos{left (32 right )} – cos{left (35 right )}end{matrix}right] right )} = sin{left (35 right )} cos{left (32 right )} – sin{left (35 right )} – sin{left (32 right )} cos{left (35 right )} + sin{left (32 right )}$$
, то
Корень xi получается делением определителя матрицы Ai. на определитель матрицы A.
( Ai получаем заменой в матрице A i-го столбца на столбец B )
$$x_{1} = frac{{det}{left (left[begin{matrix}3623 & – sin{left (32 right )} + sin{left (35 right )}370 & cos{left (32 right )} – cos{left (35 right )}end{matrix}right] right )}}{sin{left (35 right )} cos{left (32 right )} – sin{left (35 right )} – sin{left (32 right )} cos{left (35 right )} + sin{left (32 right )}} = frac{1}{sin{left (35 right )}} left(- frac{left(370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}right) left(- sin{left (32 right )} + sin{left (35 right )}right)}{- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}} + 3623right)$$
=
$$frac{- 370 sin{left (35 right )} + 370 sin{left (32 right )} + 3623 cos{left (32 right )} – 3623 cos{left (35 right )}}{sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}}$$
$$x_{2} = frac{{det}{left (left[begin{matrix}sin{left (35 right )} & 3623 – cos{left (35 right )} + 1 & 370end{matrix}right] right )}}{sin{left (35 right )} cos{left (32 right )} – sin{left (35 right )} – sin{left (32 right )} cos{left (35 right )} + sin{left (32 right )}} = frac{370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}}{- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}}$$
=
$$frac{-3623 + 3623 cos{left (35 right )} + 370 sin{left (35 right )}}{sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}}$$

Метод Гаусса
Дана система ур-ний
$$x_{1} sin{left (35 right )} + x_{2} left(- sin{left (32 right )} + sin{left (35 right )}right) = 3623$$
$$x_{1} left(- cos{left (35 right )} + 1right) + x_{2} left(cos{left (32 right )} – cos{left (35 right )}right) = 370$$

Приведём систему ур-ний к каноническому виду
$$x_{1} sin{left (35 right )} – x_{2} sin{left (32 right )} + x_{2} sin{left (35 right )} – 3623 = 0$$
$$- x_{1} cos{left (35 right )} + x_{1} + x_{2} cos{left (32 right )} – x_{2} cos{left (35 right )} – 370 = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}sin{left (35 right )} & – sin{left (32 right )} + sin{left (35 right )} & 3623 – cos{left (35 right )} + 1 & cos{left (32 right )} – cos{left (35 right )} & 370end{matrix}right]$$
В 1 ом столбце
$$left[begin{matrix}sin{left (35 right )} – cos{left (35 right )} + 1end{matrix}right]$$
делаем так, чтобы все элементы, кроме
1 го элемента равнялись нулю.
– Для этого берём 1 ую строку
$$left[begin{matrix}sin{left (35 right )} & – sin{left (32 right )} + sin{left (35 right )} & 3623end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 2 ой строки вычитаем:
$$left[begin{matrix}- – cos{left (35 right )} + 1 + – cos{left (35 right )} + 1 & – frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )} & 370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}end{matrix}right] = left[begin{matrix}0 & – frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )} & 370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}end{matrix}right]$$
получаем
$$left[begin{matrix}sin{left (35 right )} & – sin{left (32 right )} + sin{left (35 right )} & 3623 & – frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )} & 370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}end{matrix}right]$$
Во 2 ом столбце
$$left[begin{matrix}- sin{left (32 right )} + sin{left (35 right )} – frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
2 го элемента равнялись нулю.
– Для этого берём 2 ую строку
$$left[begin{matrix}0 & – frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )} & 370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 1 ой строки вычитаем:
$$left[begin{matrix}sin{left (35 right )} – 0 & – sin{left (32 right )} + sin{left (35 right )} – – sin{left (32 right )} + sin{left (35 right )} & – frac{left(370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}right) left(- sin{left (32 right )} + sin{left (35 right )}right)}{- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}} + 3623end{matrix}right] = left[begin{matrix}sin{left (35 right )} & 0 & – frac{left(370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}right) left(- sin{left (32 right )} + sin{left (35 right )}right)}{- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}} + 3623end{matrix}right]$$
получаем
$$left[begin{matrix}sin{left (35 right )} & 0 & – frac{left(370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}right) left(- sin{left (32 right )} + sin{left (35 right )}right)}{- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}} + 3623 & – frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )} & 370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}end{matrix}right]$$

Все почти готово – осталось только найти неизвестные, решая элементарные ур-ния:
$$x_{1} sin{left (35 right )} – 3623 + frac{left(370 – frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}}right) left(- sin{left (32 right )} + sin{left (35 right )}right)}{- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}} = 0$$
$$x_{2} left(- frac{1}{sin{left (35 right )}} left(- sin{left (32 right )} + sin{left (35 right )}right) left(- cos{left (35 right )} + 1right) + cos{left (32 right )} – cos{left (35 right )}right) + frac{- 3623 cos{left (35 right )} + 3623}{sin{left (35 right )}} – 370 = 0$$
Получаем ответ:
$$x_{1} = frac{- 370 sin{left (35 right )} + 370 sin{left (32 right )} + 3623 cos{left (32 right )} – 3623 cos{left (35 right )}}{sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}}$$
$$x_{2} = frac{-3623 + 3623 cos{left (35 right )} + 370 sin{left (35 right )}}{sin{left (3 right )} – sin{left (35 right )} + sin{left (32 right )}}$$

Численный ответ

x11 = 5941.59821161297
x21 = -6295.457856416895

   
4.33
Hardan
Учусь в Волгоградском Техническом Университете. Рефераты,курсовые,статьи, контрольные и др. выполняю уже в течении 4-х лет.