1)
общий алгоритм для 1(в) и 1(г) :
применяем теорему синусов.
находим 3 угол(из 180 вычитаем 2 других)
еще раз применяем теорему синусов для другой стороны
в)
![\frac{b}{sin(\beta)} = \frac{a}{sin(\alpha)} \\\alpha=180-\beta-\gamma =180-120-40=20^{\circ} \\ \frac{b}{sin(40^{\circ})} = \frac{35}{sin(20^{\circ})} \\b= \frac{sin(40^{\circ})*35}{sin(20^{\circ})} \approx 65,78 \\ \frac{a}{sin(\alpha)} = \frac{c}{sin(\gamma)} \\ \frac{35}{sin(20^{\circ})} = \frac{c}{sin(120^{\circ})} \\c= \frac{35*sin(120^{\circ})}{sin(20^{\circ})} \approx 88,62](https://tex.z-dn.net/?f=+%5Cfrac%7Bb%7D%7Bsin%28%5Cbeta%29%7D+%3D+%5Cfrac%7Ba%7D%7Bsin%28%5Calpha%29%7D+%0A%5C%5C%5Calpha%3D180-%5Cbeta-%5Cgamma+%3D180-120-40%3D20%5E%7B%5Ccirc%7D%0A%5C%5C+%5Cfrac%7Bb%7D%7Bsin%2840%5E%7B%5Ccirc%7D%29%7D+%3D+%5Cfrac%7B35%7D%7Bsin%2820%5E%7B%5Ccirc%7D%29%7D+%0A%5C%5Cb%3D+%5Cfrac%7Bsin%2840%5E%7B%5Ccirc%7D%29%2A35%7D%7Bsin%2820%5E%7B%5Ccirc%7D%29%7D+%5Capprox+65%2C78%0A%5C%5C+%5Cfrac%7Ba%7D%7Bsin%28%5Calpha%29%7D+%3D+%5Cfrac%7Bc%7D%7Bsin%28%5Cgamma%29%7D+%0A%5C%5C+%5Cfrac%7B35%7D%7Bsin%2820%5E%7B%5Ccirc%7D%29%7D+%3D+%5Cfrac%7Bc%7D%7Bsin%28120%5E%7B%5Ccirc%7D%29%7D+%0A%5C%5Cc%3D+%5Cfrac%7B35%2Asin%28120%5E%7B%5Ccirc%7D%29%7D%7Bsin%2820%5E%7B%5Ccirc%7D%29%7D+%5Capprox+88%2C62)
Ответ: b=65,87; c=88,62; α=20°
г)
![\frac{b}{sin(\gamma)} = \frac{a}{sin(\alpha)} \\ \frac{12}{sin(\gamma)} = \frac{a}{sin(36^{\circ})} \\\gamma=180-\alpha-\beta=180-36-25=119^{\circ} \\a= \frac{12*sin(36^{\circ})}{sin(119^{\circ})} \approx 8,06 \\ \frac{c}{sin(\beta)} = \frac{b}{sin(\gamma)} \\ \frac{c}{sin(25^{\circ})} = \frac{12}{sin(119^{\circ})} \\c= \frac{12*sin(25^{\circ})}{sin(119^{\circ})} \approx 5,8](https://tex.z-dn.net/?f=+%5Cfrac%7Bb%7D%7Bsin%28%5Cgamma%29%7D+%3D+%5Cfrac%7Ba%7D%7Bsin%28%5Calpha%29%7D+%0A%5C%5C+%5Cfrac%7B12%7D%7Bsin%28%5Cgamma%29%7D+%3D+%5Cfrac%7Ba%7D%7Bsin%2836%5E%7B%5Ccirc%7D%29%7D+%0A%5C%5C%5Cgamma%3D180-%5Calpha-%5Cbeta%3D180-36-25%3D119%5E%7B%5Ccirc%7D%0A%5C%5Ca%3D+%5Cfrac%7B12%2Asin%2836%5E%7B%5Ccirc%7D%29%7D%7Bsin%28119%5E%7B%5Ccirc%7D%29%7D+%5Capprox+8%2C06%0A%5C%5C+%5Cfrac%7Bc%7D%7Bsin%28%5Cbeta%29%7D+%3D++%5Cfrac%7Bb%7D%7Bsin%28%5Cgamma%29%7D+%0A%5C%5C+%5Cfrac%7Bc%7D%7Bsin%2825%5E%7B%5Ccirc%7D%29%7D+%3D+%5Cfrac%7B12%7D%7Bsin%28119%5E%7B%5Ccirc%7D%29%7D+%0A%5C%5Cc%3D+%5Cfrac%7B12%2Asin%2825%5E%7B%5Ccirc%7D%29%7D%7Bsin%28119%5E%7B%5Ccirc%7D%29%7D+%5Capprox+5%2C8)
Ответ: a=8,06; c=5,8; γ=119
3)
алгоритм:
дважды используем теорему косинусов для разных сторон
находим 3 угол как 180 минус два других
a)
![4^2=2^2+3^2-2*2*3*cos(\gamma) \\12cos(\gamma)=4+9-16 \\12cos(\gamma)=-3 \\cos(\gamma)=- \frac{1}{4} \\\gamma=arccos(-\frac{1}{4} )\approx 104,5^{\circ} \\3^2=2^2+4^2-2*2*4*cos(\beta) \\16cos(\beta)=11 \\cos(\beta)= \frac{11}{16} \\\beta=arccos(\frac{11}{16} )\approx 46,57^{\circ} \\\alpha=180-\beta-\gamma=180-104,5-46,57=28,93^{\circ}](https://tex.z-dn.net/?f=4%5E2%3D2%5E2%2B3%5E2-2%2A2%2A3%2Acos%28%5Cgamma%29%0A%5C%5C12cos%28%5Cgamma%29%3D4%2B9-16%0A%5C%5C12cos%28%5Cgamma%29%3D-3%0A%5C%5Ccos%28%5Cgamma%29%3D-+%5Cfrac%7B1%7D%7B4%7D+%0A%5C%5C%5Cgamma%3Darccos%28-%5Cfrac%7B1%7D%7B4%7D+%29%5Capprox+104%2C5%5E%7B%5Ccirc%7D%0A%5C%5C3%5E2%3D2%5E2%2B4%5E2-2%2A2%2A4%2Acos%28%5Cbeta%29%0A%5C%5C16cos%28%5Cbeta%29%3D11%0A%5C%5Ccos%28%5Cbeta%29%3D+%5Cfrac%7B11%7D%7B16%7D+%0A%5C%5C%5Cbeta%3Darccos%28%5Cfrac%7B11%7D%7B16%7D+%29%5Capprox+46%2C57%5E%7B%5Ccirc%7D%0A%5C%5C%5Calpha%3D180-%5Cbeta-%5Cgamma%3D180-104%2C5-46%2C57%3D28%2C93%5E%7B%5Ccirc%7D)
Ответ: α=28,93°; β=46,57°; γ=104,5°
б)
![8^2=2^2+7^2-2*2*7*cos(\gamma) \\28cos(\gamma)=4+49-64 \28cos(\gamma)=-11 \\cos(\gamma)=- \frac{11}{28} \\\gamma=arccos(-\frac{11}{28} )=\approx 113,1^{\circ} \\2^2=7^2+8^2-2*7*8*cos(\beta) \\112cos(\beta)=109 \\\beta=arccos( \frac{109}{112} )\approx 13,29^{\circ} \\\alpha=180-\gamma-\beta=180-113,1-13,29=53,61^{\circ}](https://tex.z-dn.net/?f=8%5E2%3D2%5E2%2B7%5E2-2%2A2%2A7%2Acos%28%5Cgamma%29%0A%5C%5C28cos%28%5Cgamma%29%3D4%2B49-64%0A%5C28cos%28%5Cgamma%29%3D-11%0A%5C%5Ccos%28%5Cgamma%29%3D-+%5Cfrac%7B11%7D%7B28%7D+%0A%5C%5C%5Cgamma%3Darccos%28-%5Cfrac%7B11%7D%7B28%7D+%29%3D%5Capprox+113%2C1%5E%7B%5Ccirc%7D%0A%5C%5C2%5E2%3D7%5E2%2B8%5E2-2%2A7%2A8%2Acos%28%5Cbeta%29%0A%5C%5C112cos%28%5Cbeta%29%3D109%0A%5C%5C%5Cbeta%3Darccos%28+%5Cfrac%7B109%7D%7B112%7D+%29%5Capprox+13%2C29%5E%7B%5Ccirc%7D%0A%5C%5C%5Calpha%3D180-%5Cgamma-%5Cbeta%3D180-113%2C1-13%2C29%3D53%2C61%5E%7B%5Ccirc%7D)
Ответ: α=53,61°; β=13,29°; γ=113,1°
в)
![7^2=4^2+5^2-2*4*5*cos(\gamma) \\40cos(\gamma)=-8 \\cos(\gamma)=- \frac{1}{5} \\\gamma=arccos(- \frac{1}{5} )\approx 101,5^{\circ} \\4^2=5^2+7^2-2*5*7*cos(\alpha) \\70cos(\alpha)=58 \\\alpha=arccos( \frac{58}{70} )\approx 34,05^{\circ} \\\beta=180-\alpha-\gamma=180-34,05-101,5=44,45^{\circ}](https://tex.z-dn.net/?f=7%5E2%3D4%5E2%2B5%5E2-2%2A4%2A5%2Acos%28%5Cgamma%29%0A%5C%5C40cos%28%5Cgamma%29%3D-8%0A%5C%5Ccos%28%5Cgamma%29%3D-+%5Cfrac%7B1%7D%7B5%7D+%0A%5C%5C%5Cgamma%3Darccos%28-+%5Cfrac%7B1%7D%7B5%7D+%29%5Capprox+101%2C5%5E%7B%5Ccirc%7D%0A%5C%5C4%5E2%3D5%5E2%2B7%5E2-2%2A5%2A7%2Acos%28%5Calpha%29%0A%5C%5C70cos%28%5Calpha%29%3D58%0A%5C%5C%5Calpha%3Darccos%28+%5Cfrac%7B58%7D%7B70%7D+%29%5Capprox+34%2C05%5E%7B%5Ccirc%7D%0A%5C%5C%5Cbeta%3D180-%5Calpha-%5Cgamma%3D180-34%2C05-101%2C5%3D44%2C45%5E%7B%5Ccirc%7D)
Ответ: α=34,05°; β=44,45°; γ=101,5°
г)
![15^2=24^2+18^2-2*24*18*cos(\alpha) \\2*24*18*cos(\alpha)=675 \\cos(\alpha)= \frac{25}{32} \\\alpha=arcccos(\frac{25}{32})\approx 38,62^{\circ} \\24^2=15^2+18^2-2*18*15*cos(\beta) \\2*18*15*cos(\beta)=-27 \\cos(\beta)=- \frac{1}{20} \\\beta=arccos(-\frac{1}{20} )\approx 92,87^{\circ} \\\gamma=180-\beta-\alpha=180-92,87- 38,62=48,51^{\circ}](https://tex.z-dn.net/?f=15%5E2%3D24%5E2%2B18%5E2-2%2A24%2A18%2Acos%28%5Calpha%29%0A%5C%5C2%2A24%2A18%2Acos%28%5Calpha%29%3D675%0A%5C%5Ccos%28%5Calpha%29%3D+%5Cfrac%7B25%7D%7B32%7D%0A%5C%5C%5Calpha%3Darcccos%28%5Cfrac%7B25%7D%7B32%7D%29%5Capprox+38%2C62%5E%7B%5Ccirc%7D%0A%5C%5C24%5E2%3D15%5E2%2B18%5E2-2%2A18%2A15%2Acos%28%5Cbeta%29%0A%5C%5C2%2A18%2A15%2Acos%28%5Cbeta%29%3D-27%0A%5C%5Ccos%28%5Cbeta%29%3D-+%5Cfrac%7B1%7D%7B20%7D+%0A%5C%5C%5Cbeta%3Darccos%28-%5Cfrac%7B1%7D%7B20%7D+%29%5Capprox+92%2C87%5E%7B%5Ccirc%7D%0A%5C%5C%5Cgamma%3D180-%5Cbeta-%5Calpha%3D180-92%2C87-+38%2C62%3D48%2C51%5E%7B%5Ccirc%7D)
Ответ: α=38,62°; β=92,87°; γ=48,51°