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Eng sodda trigonometrik tenglamalar
maktab darsligi: matematika 10-sinf, trigonometrik tenglamalar
Maqsad: sin x = a, cos x = a, tg x = a tenglamalarning umumiy yechimini yozish.
Qoida va g‘oya
Umumiy yechim formulalari (n — butun son): sin x = a (|a| ≤ 1) da x = (-1)^n · arcsin a + πn; cos x = a (|a| ≤ 1) da x = ± arccos a + 2πn; tg x = a da x = arctg a + πn. |a| > 1 bo‘lsa, sin x = a va cos x = a ning yechimi yo‘q. Xususiy hollar: sin x = 0 dan x = πn; sin x = 1 dan x = π/2 + 2πn; sin x = -1 dan x = -π/2 + 2πn; cos x = 0 dan x = π/2 + πn; cos x = 1 dan x = 2πn; cos x = -1 dan x = π + 2πn. Teskari funksiyalar: arcsin(-a) = -arcsin a, arccos(-a) = π - arccos a, arctg(-a) = -arctg a. Berilgan kesmada ildizlar kerak bo‘lsa, n ga 0, 1, 2 ... qiymatlar qo‘yib tanlang.
Namunaviy masalalar
sin x = 1/2 tenglamani yeching.
Yechim: arcsin(1/2) = π/6. x = (-1)^n · π/6 + πn.
Javob: x = (-1)^n π/6 + πn
Yechim: arcsin(1/2) = π/6. x = (-1)^n · π/6 + πn.
Javob: x = (-1)^n π/6 + πn
cos x = -1/2 tenglamani yeching.
Yechim: arccos(-1/2) = π - π/3 = 2π/3. x = ± 2π/3 + 2πn.
Javob: x = ± 2π/3 + 2πn
Yechim: arccos(-1/2) = π - π/3 = 2π/3. x = ± 2π/3 + 2πn.
Javob: x = ± 2π/3 + 2πn
tg x = sqrt(3) tenglamani yeching.
Yechim: arctg sqrt(3) = π/3. x = π/3 + πn.
Javob: x = π/3 + πn
Yechim: arctg sqrt(3) = π/3. x = π/3 + πn.
Javob: x = π/3 + πn
Tipik xatolar
- cos x = a yechimida ± belgisini va 2πn ni unutish, tg da 2πn o‘rniga πn bo‘lishi kerak.
- sin x = a ning yechimida (-1)^n ni tushirib qoldirish va faqat ikki seriyadan bittasini yozish.
- sin x = 2 yoki cos x = -3 ning yechimi bor deb hisoblash.
Mashq
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cos x = sqrt(2)/2 ni yeching.
x = ± π/4 + 2πn
2
sin x = -1/2 ni yeching.
x = (-1)^(n+1) π/6 + πn
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tg x = -1 ni yeching.
x = -π/4 + πn
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cos x = 1/2 tenglamaning [0; 2π] kesmadagi ildizlari yig‘indisini toping.
2π