Trigonometrik ayniyatlar📈 Grafikda koʻrish

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

Asosiy ayniyatlar

• ⭐ $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$
• ⭐ $\sin^2\theta + \cos^2\theta \equiv 1$, bundan $\;\sin^2\theta \equiv 1 - \cos^2\theta$, $\;\cos^2\theta \equiv 1 - \sin^2\theta$
• Ayniyatni isbotlash usullari: (a) murakkabroq tomonni soddalashtirib, ikkinchi tomonga keltirish; (b) $\text{chap} - \text{o'ng} \equiv 0$ ekanini ko'rsatish.

🟡 Boshqa trigonometrik funksiyalar

$\sec\theta = \frac{1}{\cos\theta} \qquad \operatorname{cosec}\theta = \frac{1}{\sin\theta} \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

$1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \operatorname{cosec}^2\theta$

🟡 Qo'shish formulalari

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

🟡 Ikkilangan burchak

$\sin 2A = 2\sin A\cos A \qquad \tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1$

🟡 Darajani pasaytirish

$\sin^2 A = \frac{1 - \cos 2A}{2} \qquad \cos^2 A = \frac{1 + \cos 2A}{2}$

Bular $\int \sin^2 x\,dx$ kabi integrallarda kerak bo'ladi.

🟡 $R$-shakl (harmonik shakl)

$a\sin\theta + b\cos\theta = R\sin(\theta + \alpha) \qquad a\cos\theta + b\sin\theta = R\cos(\theta - \alpha)$

$R = \sqrt{a^2 + b^2}, \qquad \tan\alpha = \frac{b}{a}$

• Eng katta qiymati $R$, eng kichigi $-R$.

🔵 Uchlangan burchak

$\sin 3A = 3\sin A - 4\sin^3 A \qquad \cos 3A = 4\cos^3 A - 3\cos A$

🔵 Yig'indini ko'paytmaga aylantirish

$\sin A + \sin B = 2\sin\frac{A + B}{2}\cos\frac{A - B}{2} \qquad \sin A - \sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}$

$\cos A + \cos B = 2\cos\frac{A + B}{2}\cos\frac{A - B}{2} \qquad \cos A - \cos B = -2\sin\frac{A + B}{2}\sin\frac{A - B}{2}$

🔵 Ko'paytmani yig'indiga aylantirish

$2\sin A\cos B = \sin(A + B) + \sin(A - B)$

$2\cos A\cos B = \cos(A + B) + \cos(A - B) \qquad 2\sin A\sin B = \cos(A - B) - \cos(A + B)$

🔵 $t$-formulalar ($t = \tan\frac{\theta}{2}$)

$\sin\theta = \frac{2t}{1 + t^2} \qquad \cos\theta = \frac{1 - t^2}{1 + t^2} \qquad \tan\theta = \frac{2t}{1 - t^2}$

🟡 ($\theta$ radianda, kichik)

$\sin\theta \approx \theta \qquad \tan\theta \approx \theta \qquad \cos\theta \approx 1 - \frac{\theta^2}{2}$