Radian, yoy va sektor

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

Radian o'lchov

• 1 radian — yoy uzunligi radiusga teng bo'lgan markaziy burchak ($\approx 57.3°$).
• ⭐ $\pi \text{ radian} = 180°$. Gradusdan radianga: $\times\dfrac{\pi}{180}$; radiandan gradusga: $\times\dfrac{180}{\pi}$.

• ⭐ Yoy uzunligi: $s = r\theta$ ($\theta$ radianda)
• ⭐ Sektor yuzi: $A = \frac{1}{2}r^2\theta$ ($\theta$ radianda)

Sektor bilan bog'liq formulalar

• Sektor perimetri: $P = 2r + r\theta$
• Vatar uzunligi: $AB = 2r\sin\dfrac{\theta}{2}$
• ⭐ Segment yuzi: $A = \frac{1}{2}r^2(\theta - \sin\theta)$

🟢 Graduslarda sektor

$\text{yoy} = \frac{\theta}{360°} \cdot 2\pi r \qquad \text{sektor} = \frac{\theta}{360°} \cdot \pi r^2$

🟢 Sektor va segment

• Sektor perimetri: $P = 2r + r\theta$
• Vatar: $AB = 2r\sin\dfrac{\theta}{2}$
• Segment yuzi: $\frac{1}{2}r^2(\theta - \sin\theta)$
• Segment perimetri: $r\theta + 2r\sin\dfrac{\theta}{2}$

🔵 Burchak tezlik

$v = r\omega \qquad (\omega \text{ — rad/s})$