Trigonometrik ayniyatlar

Trigonometrik ayniyatlar – trigonometrik tenglik α burchagiga mos keluvchi barcha qiymatlar uchun ahamiyatga ega, yaʼni chap va oʻng qismi oʻzaro teng maʼnoga ega boʻlgan trigonometriyaning asosiy boʻlimi.

Asosiy Trigonometrik ayniyatlar

№	Formula	Qiymatlar sohasi
1.1	${\sin }^2\alpha +{\cos }^2\alpha =1$	$alpha$ (barcha qiymatlarda α maʼnoga ega)
1.2	${\mathrm{tg}}^2\alpha +1=\frac{1}{{\cos }^2\alpha }={\sec }^2\alpha$	$\alpha \ne \frac{\pi }{2}+\pi n$, $n\in \mathbb{Z}$
1.3	${\mathrm{ctg}}^2\alpha +1=\frac{1}{{\sin }^2\alpha }={\mathrm{cosec}}^2\alpha$	$\alpha \ne \pi n,n\in \mathbb{Z}$
1.4	$\mathrm{tg}\alpha \cdot \mathrm{ctg}\alpha =1$	$\alpha \ne \frac{\pi n}{2},n\in \mathbb{Z}$

Qoʻshish va ayrish formulasi

sin va cos qoʻshish va ayrish chizmasi.

tg va ctg qoʻshish va ayrish chizmasi.

№	qoʻshish va ayrish formulalar
2.1	$\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
2.2	$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$
2.3	$\mathrm{tg}(\alpha \pm \beta )=\frac{\mathrm{tg}\alpha \pm \mathrm{tg}\beta }{1\mp \mathrm{tg}\alpha \mathrm{tg}\beta }$
2.4	$\mathrm{ctg}(\alpha \pm \beta )=\frac{\mathrm{ctg}\alpha \mathrm{ctg}\beta \mp 1}{\mathrm{ctg}\beta \pm \mathrm{ctg}\alpha }$

Eslatma (2.3) formulasi $\frac{\pi }{2}+\pi n$, $n\in \mathbb{Z}$ da $\alpha$, $\beta$, $\alpha \pm \beta$ maʼnoga ega emas.

Ikkilangan burchak formulasi

Agar Andoza:Math=Andoza:Math boʻlsa ikkilangan burchak formulasi (2.1)—(2.4)dan kelib chiqadi.

№	Ikkilangan burchak formulasi
3.1	$\sin 2\alpha =2\sin \alpha \cos \alpha =\frac{2\mathrm{tg}\alpha }{1+{\mathrm{tg}}^2\alpha }$
3.2	$\cos 2\alpha ={\cos }^2\alpha -{\sin }^2\alpha$ $\cos 2\alpha =2{\cos }^2\alpha -1=1-2{\sin }^2\alpha$
3.3	$\mathrm{tg}2\alpha =\frac{2\mathrm{tg}\alpha }{1-{\mathrm{tg}}^2\alpha }$
3.4	$\mathrm{ctg}2\alpha =\frac{{\mathrm{ctg}}^2\alpha -1}{2\mathrm{ctg}\alpha }$

$\mathrm{tg}2\alpha$ formula uchun:

• $\alpha =\frac{\pi }{4}+\frac{\pi }{2}n,n\in \mathbb{Z},$
• $\alpha =\frac{\pi }{2}+\pi n,n\in \mathbb{Z},$

$\mathrm{ctg}2\alpha$ formula uchun: 

$\alpha =\frac{\pi }{2}+\pi n,n\in \mathbb{Z}.$.

№	Yarim burchak formulasi
3.5	$\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}$
3.6	$\cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}$
3.7	$\mathrm{tg}\frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}=\frac{\sin \alpha }{1+\cos \alpha }=\frac{1-\cos \alpha }{\sin \alpha }=\frac{-1\pm \sqrt{1+{\mathrm{tg}}^2\alpha }}{\mathrm{tg}\alpha }$
3.8	$\mathrm{ctg}\frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }}=\frac{\sin \alpha }{1-\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha }=\mathrm{ctg}\alpha \pm \sqrt{1+{\mathrm{ctg}}^2\alpha }$

Uchlangan burchak formulasi

Agar Andoza:Math=2Andoza:Math boʻlsa ikkilangan burchak formulasi (2.1)—(2.4)dan kelib chiqadi.

№	Uchlangan burchak formulasi
4.1	$\sin 3\alpha =3\sin \alpha -4{\sin }^3\alpha$
4.2	$\cos 3\alpha =4{\cos }^3\alpha -3\cos \alpha$
4.3	$\mathrm{tg}3\alpha =\frac{3\mathrm{tg}\alpha -{\mathrm{tg}}^3\alpha }{1-3{\mathrm{tg}}^2\alpha }$
4.4	$\mathrm{ctg}3\alpha =\frac{3\mathrm{ctg}\alpha -{\mathrm{ctg}}^3\alpha }{1-3{\mathrm{ctg}}^2\alpha }$

Daraja pasaytirish formulasi

Daraja pasaytirish formulasi (3.2) orqali kelib chiqadi.

№	Sinus		№	Kosinus
5.1	${\sin }^2\alpha =\frac{1-\cos 2\alpha }{2}$		5.5	${\cos }^2\alpha =\frac{1+\cos 2\alpha }{2}$
5.2	${\sin }^3\alpha =\frac{3\sin \alpha -\sin 3\alpha }{4}$		5.6	${\cos }^3\alpha =\frac{3\cos \alpha +\cos 3\alpha }{4}$
5.3	${\sin }^4\alpha =\frac{3-4\cos 2\alpha +\cos 4\alpha }{8}$		5.7	${\cos }^4\alpha =\frac{3+4\cos 2\alpha +\cos 4\alpha }{8}$
5.4	${\sin }^5\alpha =\frac{10\sin \alpha -5\sin 3\alpha +\sin 5\alpha }{16}$		5.8	${\cos }^5\alpha =\frac{10\cos \alpha +5\cos 3\alpha +\cos 5\alpha }{16}$

№	Darajalik koʻpaytma formulasi
5.9	${\sin }^2\alpha {\cos }^2\alpha =\frac{1-\cos 4\alpha }{8}$
5.10	${\sin }^3\alpha {\cos }^3\alpha =\frac{3\sin 2\alpha -\sin 6\alpha }{32}$
5.11	${\sin }^4\alpha {\cos }^4\alpha =\frac{3-4\cos 4\alpha +\cos 8\alpha }{128}$
5.12	${\sin }^5\alpha {\cos }^5\alpha =\frac{10\sin 2\alpha -5\sin 6\alpha +\sin 10\alpha }{512}$

sin va cos ning koʻpaytmasi formulasi

№	sin va cos ning koʻpaytmasi formulasi
6.1	$\sin \alpha \sin \beta =\frac{\cos (\alpha -\beta )-\cos (\alpha +\beta )}{2}$
6.2	$\sin \alpha \cos \beta =\frac{\sin (\alpha -\beta )+\sin (\alpha +\beta )}{2}$
6.3	$\cos \alpha \cos \beta =\frac{\cos (\alpha -\beta )+\cos (\alpha +\beta )}{2}$

sin va cos ning yigʻindisi formulasi

№	sin va cos ning yigʻindisi formulasi
7.1	$\sin \alpha \pm \sin \beta =2\sin \frac{\alpha \pm \beta }{2}\cos \frac{\alpha \mp \beta }{2}$
7.2	$\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$
7.3	$\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$
7.4	$\mathrm{tg}\alpha \pm \mathrm{tg}\beta =\frac{\sin (\alpha \pm \beta )}{\cos \alpha \cos \beta }$
7.5	$\mathrm{ctg}\alpha \pm \mathrm{ctg}\beta =\frac{\sin (\beta \pm \alpha )}{\sin \alpha \sin \beta }$

Trigonometiyaning yechimlari formulasi

• $\sin x=a.$

Agarda $|a|>1$ – haqiqiy yechimlari yoʻq.

Agarda $|a|⩽1$ – $x=(-1{)}^n\arcsin a+\pi n,$, $n\in \mathbb{Z}.$ da yechimga ega.

• $\cos x=a.$

Agarda $|a|>1$ – haqiqiy yechimlari yoʻq.

Agarda $|a|⩽1$ – $x=\pm \arccos a+2\pi n,n\in \mathbb{Z}.$ da yechimga ega.

• $\mathrm{tg}x=a.$ da yechimga ega.

$x=\text{arctg}a+\pi n,n\in \mathbb{Z}.$ da yechimga ega.

• $\mathrm{ctg}x=a.$

$x=\text{arcctg}a+\pi n,n\in \mathbb{Z}.$ da yechimga ega.

Qoʻshimcha formula

tg yechimga ega boʻlsa ($\alpha \ne \pi +2\pi n$)

$\sin \alpha =\frac{2\text{ctg}\frac{\alpha }{2}}{1+{\text{ctg}}^2\frac{\alpha }{2}}$	$\cos \alpha =\frac{{\text{ctg}}^2\frac{\alpha }{2}-1}{{\text{ctg}}^2\frac{\alpha }{2}+1}$
$\text{tg}\alpha =\frac{2\text{ctg}\frac{\alpha }{2}}{{\text{ctg}}^2\frac{\alpha }{2}-1}$	$\text{ctg}\alpha =\frac{{\text{ctg}}^2\frac{\alpha }{2}-1}{2\text{ctg}\frac{\alpha }{2}}$
$\sec \alpha =\frac{{\text{ctg}}^2\frac{\alpha }{2}+1}{{\text{ctg}}^2\frac{\alpha }{2}-1}$	$\csc \alpha =\frac{1+{\text{ctg}}^2\frac{\alpha }{2}}{2\text{ctg}\frac{\alpha }{2}}$

ctg yechimga ega boʻlsa ($\alpha \ne 2\pi n$):

$\sin \alpha =\frac{2\text{ctg}\frac{\alpha }{2}}{1+{\text{ctg}}^2\frac{\alpha }{2}}$	$\cos \alpha =\frac{{\text{ctg}}^2\frac{\alpha }{2}-1}{{\text{ctg}}^2\frac{\alpha }{2}+1}$
$\text{tg}\alpha =\frac{2\text{ctg}\frac{\alpha }{2}}{{\text{ctg}}^2\frac{\alpha }{2}-1}$	$\text{ctg}\alpha =\frac{{\text{ctg}}^2\frac{\alpha }{2}-1}{2\text{ctg}\frac{\alpha }{2}}$
$\sec \alpha =\frac{{\text{ctg}}^2\frac{\alpha }{2}+1}{{\text{ctg}}^2\frac{\alpha }{2}-1}$	$\csc \alpha =\frac{1+{\text{ctg}}^2\frac{\alpha }{2}}{2\text{ctg}\frac{\alpha }{2}}$

Trigonometrik hosila va integral formulasi

$f(x)$	$f^{\prime }(x)$	$\int f(x)dx$
$\sin x$	$\cos x$	$-\cos x+C$
$\cos x$	$-\sin x$	$\sin x+C$
$\tan x$	${\sec }^{2}x$	$\ln |\sec x|+C$
$\csc x$	$-\csc x\cot x$	$\ln |\csc x-\cot x|+C$
$\sec x$	$\sec x\tan x$	$\ln |\sec x+\tan x|+C$
$\cot x$	$-{\csc }^{2}x$	$\ln |\sin x|+C$

Teskari trigonometrik funksiya

$y=\arcsin x$	$\sin y=x$	$-1\le x\le 1$	$-\frac{\pi }{2}\le y\le \frac{\pi }{2}$
$y=\arccos x$	$\cos y=x$	$-1\le x\le 1$	$0\le y\le \pi$
$y=\arctan x$	$\tan y=x$	$-\mathrm{\infty }<x<\mathrm{\infty }$	$-\frac{\pi }{2}<y<\frac{\pi }{2}$
$y=\mathrm{arccot}x$	$\cot y=x$	$-\mathrm{\infty }<x<\mathrm{\infty }$	$0<y<\pi$
$y=\mathrm{arcsec}x$	$\sec y=x$	$x<-1\text{ or }x>1$	$0\le y\le \pi ,y\ne \frac{\pi }{2}$
$y=\mathrm{arccsc}x$	$\csc y=x$	$x<-1\text{ or }x>1$	$-\frac{\pi }{2}\le y\le \frac{\pi }{2},y\ne 0$

Trigonometrik Eyler formulasi

Eyler formulasi grafigi

Eyler formulasi har qanday haqiqiy x son uchun quyidagi tenglik bajarildi:

$e^{ix}=\cos x+i\sin x,$

Eyler formulasi yordamida $\sin x$ va $\cos x$ quyidagi koʻrinishga keltirsa boʻladi:

$\sin x=\frac{e^{ix}-e^{-ix}}{2i},\cos x=\frac{e^{ix}+e^{-ix}}{2}.$

Shu formula orqali

$\mathrm{tg}x=-i\frac{e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}},\mathrm{ctg}x=i\frac{e^{ix}+e^{-ix}}{e^{ix}-e^{-ix}},$

$\sec x=\frac{2}{e^{ix}+e^{-ix}},\mathrm{cosec}x=\frac{2i}{e^{ix}-e^{-ix}}.$

Shuningdek qarang

• Trigonometriya
• Trigonometrik funksiyalar
• Uchburchak
• Aylana
• Geometriya
• Burchak
• Giperbola funksiyasi
• sin integral
• cos integral
• Sinus
• Trigonometrik funksiya
• Hosila
• Integral
• Eyler formulasi