☰ Reja · Matematika

Uchburchak: Fales, Pifagor, o‘xshashlik, sinuslar va kosinuslar

27-hafta · blok 2.2 · 3 dars
3

Sinuslar va kosinuslar teoremalari, yuz formulalari

maktab darsligi: geometriya 8–9-sinf, uchburchaklar
Maqsad: Uchburchak yuzi, ichki va tashqi chizilgan aylana radiusini topish.
Qoida va g‘oya

Yuz formulalari: S = ab sin C/2; Geron: S = sqrt(p(p-a)(p-b)(p-c)), p — yarim perimetr. Ichki chizilgan aylana radiusi r = S/p, tashqi chizilgan aylana radiusi R = abc/(4S). To‘g‘ri burchakli uchburchakda R = c/2, r = (a + b - c)/2. Sinuslar teoremasi: a/sin A = b/sin B = c/sin C = 2R. Kosinuslar teoremasi: c^2 = a^2 + b^2 - 2ab cos C; burchak topish: cos C = (a^2 + b^2 - c^2)/(2ab). Agar cos C < 0 bo‘lsa, burchak o‘tmas. Qiymatlar: sin 30° = 1/2, sin 45° = sqrt(2)/2, sin 60° = sqrt(3)/2, cos 60° = 1/2, cos 120° = -1/2.

Namunaviy masalalar
Tomonlari 13, 14, 15 bo‘lgan uchburchakning yuzini va ichki chizilgan aylana radiusini toping.
Yechim: p = 21. S = sqrt(21 · 8 · 7 · 6) = sqrt(7056) = 84. r = 84/21 = 4.
Javob: S = 84, r = 4
ABC da AB = 5, BC = 8, B = 60°. AC ni toping.
Yechim: AC^2 = 25 + 64 - 2 · 5 · 8 · 1/2 = 49, AC = 7.
Javob: 7
ABC da BC = 10, A = 30°. Tashqi chizilgan aylana radiusini toping.
Yechim: 2R = BC/sin A = 10/0,5 = 20, R = 10.
Javob: 10
Tipik xatolar
  • Kosinuslar teoremasida -2bc cos A dagi minus ishorasini unutish yoki cos 120° ning ishorasini xato olish.
  • Gerondagi p ni perimetr deb olish (p — yarim perimetr).
  • Sinuslar teoremasida a/sin A = R deb yozish (2R bo‘lishi kerak).
Mashq
1
Tomonlari 5, 5, 6 bo‘lgan uchburchakning yuzini toping.
2
ABC da a = 7, b = 5, c = 3. Eng katta burchakni toping.
3
Tashqi chizilgan aylana radiusi 5, A = 30°. BC ni toping.
4
Katetlari 6 va 8 bo‘lgan to‘g‘ri burchakli uchburchakka ichki chizilgan aylana radiusini toping.