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Yig‘indi, ayirma va ikkilangan burchak formulalari
maktab darsligi: algebra 9-sinf, matematika 10-sinf, qo‘shish formulalari
Maqsad: Qo‘shish va ikkilangan burchak formulalarini tanib, ifodani bitta funksiyaga yig‘ish.
Qoida va g‘oya
Qo‘shish formulalari: sin(α ± β) = sin α cos β ± cos α sin β; cos(α ± β) = cos α cos β ∓ sin α sin β (cos da ishora teskari!); tg(α ± β) = (tg α ± tg β)/(1 ∓ tg α tg β). Ikkilangan burchak: sin 2α = 2 sin α cos α; cos 2α = cos^2 α - sin^2 α = 1 - 2 sin^2 α = 2 cos^2 α - 1; tg 2α = 2 tg α/(1 - tg^2 α); sin 2α = 2 tg α/(1 + tg^2 α); cos 2α = (1 - tg^2 α)/(1 + tg^2 α). Ko‘p masalada ifoda formulaning o‘ng tomoniga o‘xshaydi — uni chap tomonga «yig‘ing». Masalan, cos 80° cos 20° + sin 80° sin 20° = cos(80° - 20°) = cos 60°. Burchaklar 15°, 75°, 105° bo‘lsa, ularni 45° ± 30° yoki 60° ± 45° ko‘rinishda yozing.
Namunaviy masalalar
cos 80° cos 20° + sin 80° sin 20° ni hisoblang.
Yechim: Bu cos(80° - 20°) = cos 60° = 1/2.
Javob: 1/2
Yechim: Bu cos(80° - 20°) = cos 60° = 1/2.
Javob: 1/2
sin 105° cos 15° - cos 105° sin 15° ni hisoblang.
Yechim: sin(105° - 15°) = sin 90° = 1.
Javob: 1
Yechim: sin(105° - 15°) = sin 90° = 1.
Javob: 1
sin α = 3/5, α — I chorak burchagi. sin 2α ni toping.
Yechim: cos α = 4/5. sin 2α = 2 · 3/5 · 4/5 = 24/25.
Javob: 24/25
Yechim: cos α = 4/5. sin 2α = 2 · 3/5 · 4/5 = 24/25.
Javob: 24/25
Tipik xatolar
- cos(α + β) formulasida ishorani saqlab qo‘yish: cos(α + β) = cos α cos β - sin α sin β.
- cos 2α = cos α - sin α deb yozish: ko‘rsatkichlar kvadrat.
- sin 2α = 2 sin α deb yozish: 2 sin α cos α bo‘lishi kerak.
Mashq
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cos 50° cos 10° - sin 50° sin 10° ni hisoblang.
1/2
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sin 15° cos 15° ni hisoblang.
1/4
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cos α = 3/5 bo‘lsa, cos 2α ni toping.
-7/25
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tg α = 1/2 bo‘lsa, tg(45° + α) ni toping.
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