Sinuslar va kosinuslar teoremasi

⭐ — imtihonda eng ko'p ishlatiladigan formulalar. · 🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

Uchburchak formulalari

• ⭐ Sinuslar teoremasi: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
• ⭐ Kosinuslar teoremasi: $a^2 = b^2 + c^2 - 2bc\cos A$
• ⭐ Uchburchak yuzi: $S = \frac{1}{2}ab\sin C$
• Segment yuzi = sektor yuzi − uchburchak yuzi

🟢 Kosinuslar teoremasidan burchak

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

$\cos A < 0$ bo'lsa, $A$ — o'tmas burchak.

🟢 Sinuslar teoremasining noaniq holati

$a$, $b$ va $A$ (o'tkir) berilsa:

🔵 Kengaytirilgan sinuslar teoremasi

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \qquad (R \text{ — tashqi aylana radiusi})$

🔵 Proyeksiya formulasi

$a = b\cos C + c\cos B$