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.

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

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

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

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

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

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

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

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

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

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

Agar β=2α boʻlsa ikkilangan burchak formulasi (2.1)—(2.4)dan kelib chiqadi.

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

Daraja pasaytirish formulasi (3.2) orqali kelib chiqadi.

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

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

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

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

• ${\displaystyle \sin x=a.}$

Agarda ${\displaystyle |a|>1}$ – haqiqiy yechimlari yoʻq.

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

• ${\displaystyle \cos x=a.}$

Agarda ${\displaystyle |a|>1}$ – haqiqiy yechimlari yoʻq.

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

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

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

• ${\displaystyle \operatorname {ctg} \,x=a.}$

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

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

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

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

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

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

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

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

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

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

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

Shu formula orqali

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

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

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