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
sin
2
α
+
cos
2
α
=
1
{\displaystyle \operatorname {sin} ^{2}\alpha +\operatorname {cos} ^{2}\alpha =1}
a
l
p
h
a
{\displaystyle alpha}
α maʼnoga ega)
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 {\pi }{2}}+\pi n}
n
∈
Z
{\displaystyle n\in \mathbb {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 }
α
≠
π
n
,
n
∈
Z
{\displaystyle \alpha \neq \pi n,\quad n\in \mathbb {Z} }
1.4
tg
α
⋅
ctg
α
=
1
{\displaystyle \operatorname {tg} \alpha \cdot \operatorname {ctg} \alpha =1}
α
≠
π
n
2
,
n
∈
Z
{\displaystyle \alpha \neq {\frac {\pi n}{2}},\quad n\in \mathbb {Z} }
Agar β =α boʻlsa ikkilangan burchak formulasi (2.1) —(2.4) dan kelib chiqadi.
№
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 }
α
≠
π
4
+
π
2
n
,
n
∈
Z
,
{\displaystyle \alpha \not ={\frac {\pi }{4}}+{\frac {\pi }{2}}n,n\in \mathbb {Z} ,}
α
≠
π
2
+
π
n
,
n
∈
Z
,
{\displaystyle \alpha \not ={\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} ,}
ctg
2
α
{\displaystyle \operatorname {ctg} 2\alpha }
α
≠
π
2
+
π
n
,
n
∈
Z
.
{\displaystyle \alpha \not ={\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} .}
№
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 }}}
№
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}}}
№
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,}
n
∈
Z
.
{\displaystyle n\in \mathbb {Z} .}
cos
x
=
a
.
{\displaystyle \cos x=a.}
Agarda
|
a
|
>
1
{\displaystyle |a|>1}
Agarda
|
a
|
⩽
1
{\displaystyle |a|\leqslant 1}
x
=
±
arccos
a
+
2
π
n
,
n
∈
Z
.
{\displaystyle x=\pm \arccos a+2\pi n,~n\in \mathbb {Z} .}
tg
x
=
a
.
{\displaystyle \operatorname {tg} \,x=a.}
x
=
arctg
a
+
π
n
,
n
∈
Z
.
{\displaystyle x={\text{arctg}}~a+\pi n,~n\in \mathbb {Z} .}
ctg
x
=
a
.
{\displaystyle \operatorname {ctg} \,x=a.}
x
=
arcctg
a
+
π
n
,
n
∈
Z
.
{\displaystyle x={\text{arcctg}}~a+\pi n,~n\in \mathbb {Z} .}
f
(
x
)
{\displaystyle f(x)}
f
′
(
x
)
{\displaystyle f'(x)}
∫
f
(
x
)
d
x
{\textstyle \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
tahrir
y
=
arcsin
x
{\displaystyle y=\arcsin x}
sin
y
=
x
{\displaystyle \sin y=x}
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
−
π
2
≤
y
≤
π
2
{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}
y
=
arccos
x
{\displaystyle y=\arccos x}
cos
y
=
x
{\displaystyle \cos y=x}
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
0
≤
y
≤
π
{\textstyle 0\leq y\leq \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
{\textstyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}
y
=
arccot
x
{\displaystyle y=\operatorname {arccot} x}
cot
y
=
x
{\displaystyle \cot y=x}
−
∞
<
x
<
∞
{\displaystyle -\infty <x<\infty }
0
<
y
<
π
{\textstyle 0<y<\pi }
y
=
arcsec
x
{\displaystyle y=\operatorname {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
{\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}
y
=
arccsc
x
{\displaystyle y=\operatorname {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
{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 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}}}.}
