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
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Formula
Qiymatlar sohasi
1.1
sin 2 α + cos 2 α = 1 {\displaystyle \operatorname{sin}^2 \alpha + \operatorname{cos}^2 \alpha = 1} ∀ α {\displaystyle \forall \alpha}
1.2
tg 2 α + 1 = 1 cos 2 α = sec 2 α {\displaystyle \operatorname{tg}^2 \alpha + 1 = \frac{1}{\cos^2 \alpha} = \operatorname{sec}^2 \alpha} α ≠ π 2 + π n {\displaystyle \alpha\neq\frac\pi2+\pi n} Failed to parse (Unknown function \Z): {\displaystyle n\in\Z}
1.3
ctg 2 α + 1 = 1 sin 2 α = cosec 2 α {\displaystyle \operatorname{ctg}^2 \alpha + 1 = \frac{1}{\sin^2 \alpha} = \operatorname{cosec}^2 \alpha} Failed to parse (Unknown function \Z): {\displaystyle \alpha\neq\pi n,\quad n\in\Z}
1.4
tg α ⋅ ctg α = 1 {\displaystyle \operatorname{tg} \alpha \cdot \operatorname{ctg} \alpha = 1} Failed to parse (Unknown function \Z): {\displaystyle \alpha\neq\frac{\pi n}2,\quad n\in\Z}
sin va cos qoʻshish va ayrish chizmasi. tg va ctg qoʻshish va ayrish chizmasi.
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qoʻshish va ayrish formulalar
2.1
sin ( α ± β ) = sin α cos β ± cos α sin β {\displaystyle \sin \left( \alpha \pm \beta \right) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta}
2.2
cos ( α ± β ) = cos α cos β ∓ sin α sin β {\displaystyle \cos \left( \alpha \pm \beta \right) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta}
2.3
tg ( α ± β ) = tg α ± tg β 1 ∓ tg α tg β {\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
ctg ( α ± β ) = ctg α ctg β ∓ 1 ctg β ± ctg α {\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 π 2 + π n {\displaystyle {\pi \over 2} + \pi n} n ∈ Z {\displaystyle n \in \mathbb Z} α {\displaystyle \alpha} β {\displaystyle \beta} α ± β {\displaystyle \alpha \pm \beta}
Agar β =α boʻlsa ikkilangan burchak formulasi (2.1) —(2.4) dan kelib chiqadi.
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Ikkilangan burchak formulasi
3.1
sin 2 α = 2 sin α cos α = 2 tg α 1 + tg 2 α {\displaystyle \operatorname{sin} 2 \alpha = 2 {\sin \alpha}\ {\cos \alpha} = \frac{2\operatorname{tg} \alpha}{1+\operatorname{tg}^2 \alpha}}
3.2
cos 2 α = cos 2 α − sin 2 α {\displaystyle \operatorname{cos} 2 \alpha = {\cos^2 \alpha} - {\sin^2 \alpha}} cos 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α {\displaystyle \operatorname{cos} 2 \alpha = 2 {\cos^2 \alpha} - 1 = 1 - 2 {\sin^2 \alpha}}
3.3
tg 2 α = 2 tg α 1 − tg 2 α {\displaystyle \operatorname{tg} 2 \alpha = \frac{2 \operatorname{tg} \alpha}{1 - \operatorname{tg}^2 \alpha}}
3.4
ctg 2 α = ctg 2 α − 1 2 ctg α {\displaystyle \operatorname{ctg} 2 \alpha = \frac{\operatorname{ctg}^2 \alpha - 1}{2 \operatorname{ctg} \alpha}}
tg 2 α {\displaystyle \operatorname{tg} 2 \alpha} Failed to parse (Unknown function \not): {\displaystyle \alpha \not=\frac{\pi}4 + \frac{\pi}2 n, n \in \mathbb Z,} Failed to parse (Unknown function \not): {\displaystyle \alpha \not=\frac{\pi}2 + \pi n, n\in \mathbb Z,} ctg 2 α {\displaystyle \operatorname{ctg} 2 \alpha} Failed to parse (Unknown function \not): {\displaystyle \alpha \not=\frac{\pi}2 + \pi n, n \in \mathbb Z.} .
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Yarim burchak formulasi
3.5
sin α 2 = ± 1 − cos α 2 {\displaystyle \sin{\alpha\over 2}=\pm\sqrt{1-\cos\alpha \over 2}}
3.6
cos α 2 = ± 1 + cos α 2 {\displaystyle \cos{\alpha \over 2}=\pm\sqrt{1+\cos\alpha\over 2}}
3.7
tg α 2 = ± 1 − cos α 1 + cos α = sin α 1 + cos α = 1 − cos α sin α = − 1 ± 1 + tg 2 α tg α {\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
ctg α 2 = ± 1 + cos α 1 − cos α = sin α 1 − cos α = 1 + cos α sin α = ctg α ± 1 + ctg 2 α {\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.
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Uchlangan burchak formulasi
4.1
sin 3 α = 3 sin α − 4 sin 3 α {\displaystyle \sin 3\alpha = 3 \sin \alpha - 4 \sin^3\alpha}
4.2
cos 3 α = 4 cos 3 α − 3 cos α {\displaystyle \cos 3\alpha = 4 \cos^3\alpha - 3 \cos \alpha}
4.3
tg 3 α = 3 tg α − tg 3 α 1 − 3 tg 2 α {\displaystyle \operatorname{tg} 3\alpha = \frac{3 \operatorname{tg}\alpha - \operatorname{tg}^3\alpha}{1 - 3 \operatorname{tg}^2\alpha}}
4.4
ctg 3 α = 3 ctg α − ctg 3 α 1 − 3 ctg 2 α {\displaystyle \operatorname{ctg} 3\alpha = \frac{3 \operatorname{ctg}\alpha - \operatorname{ctg}^3\alpha}{1 - 3 \operatorname{ctg}^2\alpha}}
Eslatma
tg 3 α {\displaystyle \operatorname{tg} 3\alpha} Failed to parse (Unknown function \not): {\displaystyle \alpha \not=\frac{\pi}6 + \frac{\pi}3 n, n \in \mathbb Z} ctg 3 α {\displaystyle \operatorname{ctg} 3\alpha} Failed to parse (Unknown function \not): {\displaystyle \alpha \not=\frac{\pi}3 n + \pi n, n \in \mathbb Z} ;
Daraja pasaytirish formulasi (3.2) orqali kelib chiqadi.
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Sinus
№
Kosinus
5.1
sin 2 α = 1 − cos 2 α 2 {\displaystyle \sin^2\alpha = \frac{1 - \cos 2\alpha}{2}} 5.5
cos 2 α = 1 + cos 2 α 2 {\displaystyle \cos^2\alpha = \frac{1 + \cos 2\alpha}{2}}
5.2
sin 3 α = 3 sin α − sin 3 α 4 {\displaystyle \sin^3\alpha = \frac{3 \sin\alpha - \sin 3\alpha}{4}} 5.6
cos 3 α = 3 cos α + cos 3 α 4 {\displaystyle \cos^3\alpha = \frac{3 \cos\alpha + \cos 3\alpha}{4}}
5.3
sin 4 α = 3 − 4 cos 2 α + cos 4 α 8 {\displaystyle \sin^4\alpha = \frac{3 - 4 \cos 2\alpha + \cos 4\alpha}{8}} 5.7
cos 4 α = 3 + 4 cos 2 α + cos 4 α 8 {\displaystyle \cos^4\alpha = \frac{3 + 4 \cos 2\alpha + \cos 4\alpha}{8}}
5.4
sin 5 α = 10 sin α − 5 sin 3 α + sin 5 α 16 {\displaystyle \sin^5\alpha = \frac{10 \sin\alpha - 5 \sin 3\alpha + \sin 5\alpha}{16}} 5.8
cos 5 α = 10 cos α + 5 cos 3 α + cos 5 α 16 {\displaystyle \cos^5\alpha = \frac{10 \cos\alpha + 5 \cos 3\alpha + \cos 5\alpha}{16}}
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Darajalik koʻpaytma formulasi
5.9
sin 2 α cos 2 α = 1 − cos 4 α 8 {\displaystyle \sin^2\alpha \cos^2\alpha = \frac{1 - \cos 4\alpha}{8}}
5.10
sin 3 α cos 3 α = 3 sin 2 α − sin 6 α 32 {\displaystyle \sin^3\alpha \cos^3\alpha = \frac{3\sin 2\alpha - \sin 6\alpha}{32}}
5.11
sin 4 α cos 4 α = 3 − 4 cos 4 α + cos 8 α 128 {\displaystyle \sin^4\alpha \cos^4\alpha = \frac{3-4\cos 4\alpha + \cos 8\alpha}{128}}
5.12
sin 5 α cos 5 α = 10 sin 2 α − 5 sin 6 α + sin 10 α 512 {\displaystyle \sin^5\alpha \cos^5\alpha = \frac{10\sin 2\alpha - 5\sin 6\alpha + \sin 10\alpha}{512}}
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sin va cos ning koʻpaytmasi formulasi
6.1
sin α sin β = cos ( α − β ) − cos ( α + β ) 2 {\displaystyle \sin \alpha \sin \beta = \frac{ \cos ( \alpha - \beta) - \cos ( \alpha + \beta)}{2}}
6.2
sin α cos β = sin ( α − β ) + sin ( α + β ) 2 {\displaystyle \sin \alpha \cos \beta = \frac{ \sin ( \alpha - \beta) + \sin ( \alpha + \beta) }{2}}
6.3
cos α cos β = cos ( α − β ) + cos ( α + β ) 2 {\displaystyle \cos \alpha \cos \beta = \frac{ \cos ( \alpha - \beta) + \cos ( \alpha + \beta)}{2}}
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sin va cos ning yigʻindisi formulasi
7.1
sin α ± sin β = 2 sin α ± β 2 cos α ∓ β 2 {\displaystyle \sin \alpha \pm \sin \beta = 2 \sin \frac{ \alpha \pm \beta}{2} \cos \frac{ \alpha \mp \beta}{2}}
7.2
cos α + cos β = 2 cos α + β 2 cos α − β 2 {\displaystyle \cos \alpha + \cos \beta = 2 \cos \frac{ \alpha + \beta}{2} \cos \frac{ \alpha - \beta}{2}}
7.3
cos α − cos β = − 2 sin α + β 2 sin α − β 2 {\displaystyle \cos \alpha - \cos \beta = - 2 \sin \frac{ \alpha + \beta}{2} \sin \frac{ \alpha - \beta}{2}}
7.4
tg α ± tg β = sin ( α ± β ) cos α cos β {\displaystyle \operatorname{tg} \alpha \pm \operatorname{tg} \beta = \frac{ \sin ( \alpha \pm \beta)}{ \cos \alpha \cos \beta}}
7.5
ctg α ± ctg β = sin ( β ± α ) sin α sin β {\displaystyle \operatorname{ctg} \alpha \pm \operatorname{ctg} \beta = \frac{ \sin ( \beta \pm \alpha)}{ \sin \alpha \sin \beta}}
sin x = a . {\displaystyle \sin x = a.} Agarda | a | > 1 {\displaystyle |a|>1}
Agarda | a | ⩽ 1 {\displaystyle |a| \leqslant 1} x = ( − 1 ) n arcsin a + π n , {\displaystyle x=(-1)^n\arcsin a+\pi n,} Failed to parse (Unknown function \Z): {\displaystyle n\in\Z.} da yechimga ega. cos x = a . {\displaystyle \cos x = a.} Agarda | a | > 1 {\displaystyle |a|>1}
Agarda | a | ⩽ 1 {\displaystyle |a| \leqslant 1} Failed to parse (Unknown function \Z): {\displaystyle x=\pm\arccos a+2\pi n,~n\in\Z.} da yechimga ega. tg x = a . {\displaystyle \operatorname{tg}\, x = a.} Failed to parse (Unknown function \Z): {\displaystyle x=\text{arctg}~a+\pi n,~n\in\Z.} da yechimga ega.ctg x = a . {\displaystyle \operatorname{ctg}\, x = a.} Failed to parse (Unknown function \Z): {\displaystyle x=\text{arcctg}~a+\pi n,~n\in\Z.} da yechimga ega.
tg yechimga ega boʻlsa (α ≠ π + 2 π n {\displaystyle \alpha\ne\pi+2\pi n}
sin α = 2 ctg α 2 1 + ctg 2 α 2 {\displaystyle \sin\alpha =\frac{2~\text{ctg}~\frac\alpha2} {1+\text{ctg}^2\frac\alpha2}} cos α = ctg 2 α 2 − 1 ctg 2 α 2 + 1 {\displaystyle \cos\alpha = \frac{\text{ctg}^2\frac\alpha2-1}{\text{ctg}^2\frac\alpha2+1}}
tg α = 2 ctg α 2 ctg 2 α 2 − 1 {\displaystyle \text{tg}~\alpha=\frac{2~\text{ctg}~\frac\alpha2}{\text{ctg}^2\frac\alpha2-1}} ctg α = ctg 2 α 2 − 1 2 ctg α 2 {\displaystyle \text{ctg}~\alpha=\frac{\text{ctg}^2\frac\alpha2-1}{2~\text{ctg}~\frac\alpha2}}
sec α = ctg 2 α 2 + 1 ctg 2 α 2 − 1 {\displaystyle \sec\alpha=\frac{\text{ctg}^2\frac\alpha2+1}{\text{ctg}^2\frac\alpha2-1}} csc α = 1 + ctg 2 α 2 2 ctg α 2 {\displaystyle \csc\alpha =\frac{1+\text{ctg}^2\frac\alpha2}{2~\text{ctg}~\frac\alpha2}}
ctg yechimga ega boʻlsa (α ≠ 2 π n {\displaystyle \alpha\ne2\pi n}
sin α = 2 ctg α 2 1 + ctg 2 α 2 {\displaystyle \sin\alpha =\frac{2~\text{ctg}~\frac\alpha2} {1+\text{ctg}^2\frac\alpha2}} cos α = ctg 2 α 2 − 1 ctg 2 α 2 + 1 {\displaystyle \cos\alpha = \frac{\text{ctg}^2\frac\alpha2-1}{\text{ctg}^2\frac\alpha2+1}}
tg α = 2 ctg α 2 ctg 2 α 2 − 1 {\displaystyle \text{tg}~\alpha=\frac{2~\text{ctg}~\frac\alpha2}{\text{ctg}^2\frac\alpha2-1}} ctg α = ctg 2 α 2 − 1 2 ctg α 2 {\displaystyle \text{ctg}~\alpha=\frac{\text{ctg}^2\frac\alpha2-1}{2~\text{ctg}~\frac\alpha2}}
sec α = ctg 2 α 2 + 1 ctg 2 α 2 − 1 {\displaystyle \sec\alpha=\frac{\text{ctg}^2\frac\alpha2+1}{\text{ctg}^2\frac\alpha2-1}} csc α = 1 + ctg 2 α 2 2 ctg α 2 {\displaystyle \csc\alpha =\frac{1+\text{ctg}^2\frac\alpha2}{2~\text{ctg}~\frac\alpha2}}
f ( x ) {\displaystyle f(x)} f ′ ( x ) {\displaystyle f'(x)} ∫ f ( x ) d x {\displaystyle \int f(x) \, dx}
sin x {\displaystyle \sin x} cos x {\displaystyle \cos x} − cos x + C {\displaystyle -\cos x + C}
cos x {\displaystyle \cos x} − sin x {\displaystyle -\sin x} sin x + C {\displaystyle \sin x + C}
tan x {\displaystyle \tan x} sec 2 x {\displaystyle \sec^2 x} ln | sec x | + C {\displaystyle \ln \left| \sec x \right| + C}
csc x {\displaystyle \csc x} − csc x cot x {\displaystyle -\csc x \cot x} ln | csc x − cot x | + C {\displaystyle \ln \left| \csc x - \cot x \right| + C}
sec x {\displaystyle \sec x} sec x tan x {\displaystyle \sec x \tan x} ln | sec x + tan x | + C {\displaystyle \ln \left| \sec x + \tan x \right| + C}
cot x {\displaystyle \cot x} − csc 2 x {\displaystyle -\csc^2 x} ln | sin x | + C {\displaystyle \ln \left| \sin x \right| + C}
Teskari trigonometrik funksiya
y = arcsin x {\displaystyle y = \arcsin x} sin y = x {\displaystyle \sin y = x} − 1 ≤ x ≤ 1 {\displaystyle -1 \le x \le 1} − π 2 ≤ y ≤ π 2 {\displaystyle -\frac{\pi}{2} \le y \le \frac{\pi}{2}}
y = arccos x {\displaystyle y = \arccos x} cos y = x {\displaystyle \cos y = x} − 1 ≤ x ≤ 1 {\displaystyle -1 \le x \le 1} 0 ≤ y ≤ π {\displaystyle 0 \le y \le \pi}
y = arctan x {\displaystyle y = \arctan x} tan y = x {\displaystyle \tan y = x} − ∞ < x < ∞ {\displaystyle -\infty < x < \infty} − π 2 < y < π 2 {\displaystyle -\frac{\pi}{2} < y < \frac{\pi}{2}}
y = arccot x {\displaystyle y = \arccot x} cot y = x {\displaystyle \cot y = x} − ∞ < x < ∞ {\displaystyle -\infty < x < \infty} 0 < y < π {\displaystyle 0 < y < \pi}
y = arcsec x {\displaystyle y = \arcsec x} sec y = x {\displaystyle \sec y = x} x < − 1 or x > 1 {\displaystyle x<-1 \text{ or } x>1} 0 ≤ y ≤ π , y ≠ π 2 {\displaystyle 0 \le y \le \pi,\; y \ne \frac{\pi}{2}}
y = arccsc x {\displaystyle y = \arccsc x} csc y = x {\displaystyle \csc y = x} x < − 1 or x > 1 {\displaystyle x<-1 \text{ or } x>1} − π 2 ≤ y ≤ π 2 , y ≠ 0 {\displaystyle -\frac{\pi}{2} \le y \le \frac{\pi}{2},\; y \ne 0}
Eyler formulasi grafigi Eyler formulasi har qanday haqiqiy x son uchun quyidagi tenglik bajarildi:
e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} Eyler formulasi yordamida sin x {\displaystyle \sin x} cos x {\displaystyle \cos x}
sin x = e i x − e − i x 2 i , cos x = e i x + e − i x 2 . {\displaystyle \sin x=\frac{e^{ix}-e^{-ix}}{2i}, \qquad \qquad \cos x=\frac{e^{ix}+e^{-ix}}{2}.} Shu formula orqali
tg x = − i e i x − e − i x e i x + e − i x , ctg x = i e i x + e − i x e i x − e − i x , {\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}},} sec x = 2 e i x + e − i x , cosec x = 2 i e i x − e − i x . {\displaystyle \sec x = \frac{2}{e^{ix}+e^{-ix}}, \qquad \qquad \operatorname{cosec}\, x=\frac{2i}{e^{ix}-e^{-ix}}.}
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