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	$\operatorname {sin} ^{2}\alpha +\operatorname {cos} ^{2}\alpha =1$	$alpha$ (barcha qiymatlarda α maʼnoga ega)
1.2	$\operatorname {tg} ^{2}\alpha +1={\frac {1}{\cos ^{2}\alpha }}=\operatorname {sec} ^{2}\alpha$	$\alpha \neq {\frac {\pi }{2}}+\pi n$, $n\in \mathbb {Z}$
1.3	$\operatorname {ctg} ^{2}\alpha +1={\frac {1}{\sin ^{2}\alpha }}=\operatorname {cosec} ^{2}\alpha$	$\alpha \neq \pi n,\quad n\in \mathbb {Z}$
1.4	$\operatorname {tg} \alpha \cdot \operatorname {ctg} \alpha =1$	$\alpha \neq {\frac {\pi n}{2}},\quad n\in \mathbb {Z}$

Qoʻshish va ayrish formulasi

sin va cos qoʻshish va ayrish chizmasi.

tg va ctg qoʻshish va ayrish chizmasi.

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

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

Ikkilangan burchak formulasi

Agar β=α boʻlsa ikkilangan burchak formulasi (2.1)—(2.4)dan kelib chiqadi.

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

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

• $\alpha \not ={\frac {\pi }{4}}+{\frac {\pi }{2}}n,n\in \mathbb {Z} ,$
• $\alpha \not ={\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} ,$

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

$\alpha \not ={\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} .$.

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

Uchlangan burchak formulasi

Agar β=2α 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	$\operatorname {tg} 3\alpha ={\frac {3\operatorname {tg} \alpha -\operatorname {tg} ^{3}\alpha }{1-3\operatorname {tg} ^{2}\alpha }}$
4.4	$\operatorname {ctg} 3\alpha ={\frac {3\operatorname {ctg} \alpha -\operatorname {ctg} ^{3}\alpha }{1-3\operatorname {ctg} ^{2}\alpha }}$

Eslatma  

$\operatorname {tg} 3\alpha$ formula uchun: $\alpha \not ={\frac {\pi }{6}}+{\frac {\pi }{3}}n,n\in \mathbb {Z}$
$\operatorname {ctg} 3\alpha$ formula uchun: $\alpha \not ={\frac {\pi }{3}}n+\pi n,n\in \mathbb {Z}$;

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	$\operatorname {tg} \alpha \pm \operatorname {tg} \beta ={\frac {\sin(\alpha \pm \beta )}{\cos \alpha \cos \beta }}$
7.5	$\operatorname {ctg} \alpha \pm \operatorname {ctg} \beta ={\frac {\sin(\beta \pm \alpha )}{\sin \alpha \sin \beta }}$

Trigonometiyaning yechimlari formulasi

• $\sin x=a.$

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

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

• $\cos x=a.$

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

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

• $\operatorname {tg} \,x=a.$ da yechimga ega.

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

• $\operatorname {ctg} \,x=a.$

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

Qoʻshimcha formula

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

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

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

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

Trigonometrik hosila va integral formulasi

$f(x)$	$f'(x)$	${\textstyle \int f(x)\,dx}$
$\sin x$	$\cos x$	$-\cos x+C$
$\cos x$	$-\sin x$	$\sin x+C$
$\tan x$	$\sec ^{2}x$	$\ln \left|\sec x\right|+C$
$\csc x$	$-\csc x\cot x$	$\ln \left|\csc x-\cot x\right|+C$
$\sec x$	$\sec x\tan x$	$\ln \left|\sec x+\tan x\right|+C$
$\cot x$	$-\csc ^{2}x$	$\ln \left|\sin x\right|+C$

Teskari trigonometrik funksiya

$y=\arcsin x$	$\sin y=x$	$-1\leq x\leq 1$	${\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}$
$y=\arccos x$	$\cos y=x$	$-1\leq x\leq 1$	${\textstyle 0\leq y\leq \pi }$
$y=\arctan x$	$\tan y=x$	$-\infty <x<\infty$	${\textstyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}$
$y=\operatorname {arccot} x$	$\cot y=x$	$-\infty <x<\infty$	${\textstyle 0<y<\pi }$
$y=\operatorname {arcsec} x$	$\sec y=x$	$x<-1{\text{ or }}x>1$	${\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}$
$y=\operatorname {arccsc} x$	$\csc y=x$	$x<-1{\text{ or }}x>1$	${\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}$

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:

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

Shu formula orqali

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

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

Shuningdek qarang

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