1.3 Trigonometriya — Nazariya

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Kirish

Trigonometriya — burchaklar va uchburchaklar bilan ishlaydigan matematika bo'limi. Robotika va dronlarda:

• 🤖 Robot qo'llari — bo'g'in burchaklari
• 🚁 Dron yo'nalishi — Euler burchaklari (roll, pitch, yaw)
• 🚀 Raketa navigatsiyasi — burchak hisoblash
• 📐 Sensor kalibrlash — burchak o'zgarishlari

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1. Asosiy Trigonometrik Funksiyalar

To'g'ri burchakli uchburchakda

        /|
       / |
    c /  | a (qarama-qarshi)
     /   |
    /θ___|
       b (yonidagi)

$$\sin\theta = \frac{a}{c} = \frac{\text{qarama-qarshi}}{\text{gipotenuza}}$$ $$\cos\theta = \frac{b}{c} = \frac{\text{yonidagi}}{\text{gipotenuza}}$$ $$\tan\theta = \frac{a}{b} = \frac{\sin\theta}{\cos\theta}$$

Qo'shimcha funksiyalar

$$\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$$ $$\sec\theta = \frac{1}{\cos\theta}$$ $$\csc\theta = \frac{1}{\sin\theta}$$

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2. Birlik Aylana

Radius 1 bo'lgan aylana:

$$x = \cos\theta, \quad y = \sin\theta$$

Asosiy qiymatlar

θ (gradus)	θ (radian)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	∞
180°	π	0	-1	0
270°	3π/2	-1	0	∞
360°	2π	0	1	0

Radian va Gradus

$$\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$$ $$\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}$$

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3. Asosiy Trigonometrik Tengliklar

Pifagor tengligi

$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$

Juftlik xossalari

$$\sin(-\theta) = -\sin\theta \quad \text{(toq funksiya)}$$ $$\cos(-\theta) = \cos\theta \quad \text{(juft funksiya)}$$ $$\tan(-\theta) = -\tan\theta \quad \text{(toq funksiya)}$$

Ko'shimcha burchak

$$\sin(90° - \theta) = \cos\theta$$ $$\cos(90° - \theta) = \sin\theta$$ $$\tan(90° - \theta) = \cot\theta$$

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4. Yig'indi va Ayirma Formulalari

Sinuslar

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$ $$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$$

Kosinuslar

$$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$$ $$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$$

Tangenslar

$$\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$$ $$\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}$$

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5. Ikkilangan Burchak Formulalari

$$\sin(2\theta) = 2\sin\theta\cos\theta$$ $$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$ $$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$

Yarim burchak

$$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$ $$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$

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6. Uchburchak Teoremalari

Sinus Teoremasi

Ixtiyoriy uchburchak uchun:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

Bu yerda $R$ — tashqi aylana radiusi.

Qo'llanilish: Ikkita burchak va bitta tomon ma'lum bo'lganda.

Kosinus Teoremasi

$$c^2 = a^2 + b^2 - 2ab\cos C$$ $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$

Qo'llanilish: Uchta tomon yoki ikkita tomon va orasidagi burchak ma'lum bo'lganda.

Yuza formulalari

$$S = \frac{1}{2}ab\sin C$$ $$S = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{(Geron formulasi)}$$

Bu yerda $s = \frac{a+b+c}{2}$ — yarim perimetr.

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7. Teskari Trigonometrik Funksiyalar

Ta'riflar

$$\theta = \arcsin(x) \Leftrightarrow \sin\theta = x, \quad \theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$$ $$\theta = \arccos(x) \Leftrightarrow \cos\theta = x, \quad \theta \in [0, \pi]$$ $$\theta = \arctan(x) \Leftrightarrow \tan\theta = x, \quad \theta \in (-\frac{\pi}{2}, \frac{\pi}{2})$$

atan2 funksiyasi

Robotikada eng ko'p ishlatiladigan:

$$\theta = \text{atan2}(y, x)$$

Bu funksiya to'liq $(-\pi, \pi]$ oralig'ida to'g'ri burchakni qaytaradi:

Chorak	x	y	atan2(y,x)
I	+	+	0 dan π/2
II	-	+	π/2 dan π
III	-	-	-π dan -π/2
IV	+	-	-π/2 dan 0

Muhim

Robotikada arctan(y/x) emas, atan2(y, x) ishlating!

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8. Euler Burchaklari

Ta'rif

Uch burchak orqali 3D yo'nalishni ifodalash:

• Roll (φ) — X o'qi atrofida aylanish
• Pitch (θ) — Y o'qi atrofida aylanish
• Yaw (ψ) — Z o'qi atrofida aylanish

Aylanish matritsasidan Euler burchaklariga

ZYX konvensiya (Tait-Bryan):

$$R = R_z(\psi) R_y(\theta) R_x(\phi)$$ $$R = \begin{bmatrix} c_\psi c_\theta & c_\psi s_\theta s_\phi - s_\psi c_\phi & c_\psi s_\theta c_\phi + s_\psi s_\phi \\ s_\psi c_\theta & s_\psi s_\theta s_\phi + c_\psi c_\phi & s_\psi s_\theta c_\phi - c_\psi s_\phi \\ -s_\theta & c_\theta s_\phi & c_\theta c_\phi \end{bmatrix}$$

Bu yerda $c = \cos$, $s = \sin$.

Matritsadan burchaklarni ajratib olish

$$\theta = \arcsin(-r_{31})$$ $$\phi = \text{atan2}(r_{32}, r_{33})$$ $$\psi = \text{atan2}(r_{21}, r_{11})$$

Gimbal Lock

$\theta = \pm 90°$ da gimbal lock muammosi yuzaga keladi — bir daraja erkinlik yo'qoladi.

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9. Robotikada Qo'llanilish

Robot qo'li burchaklari

Inverse kinematics — maqsad pozitsiyadan burchaklarni topish:

         θ2
    L2  /
       /
      ●
     /
    / θ1
   ●------→ x

2 bo'g'inli robot uchun:

$$x = L_1\cos\theta_1 + L_2\cos(\theta_1 + \theta_2)$$ $$y = L_1\sin\theta_1 + L_2\sin(\theta_1 + \theta_2)$$

Inverse:

$$\cos\theta_2 = \frac{x^2 + y^2 - L_1^2 - L_2^2}{2L_1L_2}$$ $$\theta_1 = \text{atan2}(y, x) - \text{atan2}(L_2\sin\theta_2, L_1 + L_2\cos\theta_2)$$

Dron pozitsiyasi

GPS koordinatalardan burchak:

$$\text{bearing} = \text{atan2}(\sin(\Delta\lambda)\cos\phi_2, \cos\phi_1\sin\phi_2 - \sin\phi_1\cos\phi_2\cos(\Delta\lambda))$$

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Xulosa

Formula	Qo'llanilish
$\sin^2 + \cos^2 = 1$	Asosiy tenglik
$\sin(\alpha+\beta)$	Burchak qo'shish
Kosinus teoremasi	Masofa hisoblash
atan2(y, x)	Burchak aniqlash
Euler burchaklari	3D yo'nalish

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