☰ Mundarija · Algebra

Ikkilangan burchak va keltirish formulalari

Darslar 26–27 · 2 ta dars · Sh.A. Alimov, A.R. Xalmuxamedov, M.A. Mirzaxmedov. Algebra: umumiy o‘rta ta’lim maktablarining 9-sinfi uchun darslik. Qayta ishlangan 4-nashri. Toshkent: «O‘qituvchi» NMIU, 2019
26

Ikkilangan burchakning sinusi va kosinusi

Darslik: 126–128-betlar
MaqsadIkkilangan burchakning sinusi, kosinusi va tangensi formulalarini bilish hamda hisoblash va soddalashtirishda qo‘llash.
Yangi so‘zlar
ikkilangan burchak · double anglesin 2α formulasi · formula for sin 2αcos 2α formulasi · formula for cos 2αtg 2α formulasi · formula for tan 2α
Tushuntirish

Qo‘shish formulalarida β = α deb olsak, ikkilangan burchak formulalari hosil bo‘ladi. sin 2α = 2 sin α cos α. cos 2α = cos²α − sin²α; asosiy ayniyat yordamida uni cos 2α = 2cos²α − 1 yoki cos 2α = 1 − 2sin²α ko‘rinishda ham yozish mumkin. tg 2α = 2tg α : (1 − tg²α), bunda tg α va tg 2α mavjud. Bu formulalar bir burchakning funksiyalari ma’lum bo‘lsa, ikki baravar burchakning funksiyalarini topishga yordam beradi, masalan sin 30° = 2 sin 15° cos 15° dan 2 sin 15° cos 15° = 1/2. Shuningdek, 1 + cos 2α = 2cos²α va 1 − cos 2α = 2sin²α ayniyatlari ifodalarni soddalashtirishda foydali. Sinus va kosinusni topishda burchak choragini hisobga olib, ildiz oldidagi ishorani to‘g‘ri tanlang.

Namunalar
sin α = 3/5, 0 < α < π/2, demak cos α = 4/5. sin 2α = 2 · (3/5) · (4/5) = 24/25; cos 2α = 1 − 2 · (9/25) = 7/25.
sin 2α : (2 cos α) = 2 sin α cos α : (2 cos α) = sin α. cos² 15° − sin² 15° = cos 30° = √3/2.
Sinfda faoliyat

«Ikki baravar»: o‘qituvchi bitta burchakning sinusi va kosinusini beradi (3/5 va 4/5; 5/13 va 12/13); jamoalar tezda ikkilangan burchakning sin 2α va cos 2α ini hisoblaydi.

Mashq
1
2 sin 15° cos 15° ni hisoblang.
2
cos α = 1/3 bo‘lsa, cos 2α ni toping.
3
tg α = 3 bo‘lsa, tg 2α ni toping.
4
cos 2α = 1 − 2sin²α ni cos 2α = cos²α − sin²α dan keltirib chiqaring.