Trigonometrijski identiteti su izrazi jednakosti koji pokazuju poveznice između pojedinih trigonometrijskih funkcija . Ti izrazi su istiniti za svaku odabranu vrijednost određene varijable (kuta ili nekog drugog broja). Kako su trigonometrijske funkcije međusobno povezane pomoću vrijednosti jedne, moguće je izraziti neku drugu funkciju . Identiteti se koriste kada je potrebno pojednostaviti izraze koji uključuju trigonometrijske funkcije.
Sinusi i kosinusi u jediničnoj kružnici Nazivlje
Kutovi Podrobniji članak o temi:
Kut Imena kutova se daju prema slovima grčkog alfabeta kao što su alfa (α ), beta (β ), gama (γ ), delta (δ ) i theta (θ ). Mjerne jedinice za mjerenje kutova su stupnjevi , radijani i gradi :
1 puni krug = 360 stupnjeva = 2
π
{\displaystyle \pi }
Slijedeća tablica prikazuje pretvorbu mjernih jedinica za određene veličine kuteva:
Stupnjevi
30°
60°
120°
150°
210°
240°
300°
330°
Radijani
π
6
{\displaystyle {\frac {\pi }{6}}\!}
π
3
{\displaystyle {\frac {\pi }{3}}\!}
2
π
3
{\displaystyle {\frac {2\pi }{3}}\!}
5
π
6
{\displaystyle {\frac {5\pi }{6}}\!}
7
π
6
{\displaystyle {\frac {7\pi }{6}}\!}
4
π
3
{\displaystyle {\frac {4\pi }{3}}\!}
5
π
3
{\displaystyle {\frac {5\pi }{3}}\!}
11
π
6
{\displaystyle {\frac {11\pi }{6}}\!}
Gradi
33⅓
66⅔
133⅓
166⅔
233⅓
266⅔
333⅓
366⅔
Stupnjevi
45°
90°
135°
180°
225°
270°
315°
360°
Radijani
π
4
{\displaystyle {\frac {\pi }{4}}\!}
π
2
{\displaystyle {\frac {\pi }{2}}\!}
3
π
4
{\displaystyle {\frac {3\pi }{4}}\!}
π
{\displaystyle \pi \!}
5
π
4
{\displaystyle {\frac {5\pi }{4}}\!}
3
π
2
{\displaystyle {\frac {3\pi }{2}}\!}
7
π
4
{\displaystyle {\frac {7\pi }{4}}\!}
2
π
{\displaystyle 2\pi \!}
Gradi
50
100
150
200
250
300
350
400
Kutovi se u trigonometriji najčešće izražavaju u radijanima i to bez mjerne jedinice, stupnjevi s oznakom ° se manje koriste, a gradi izrazito rijetko.
Trigonometrijske funkcije Primarne trigonometrijske funkcije su sinus i kosinus kuta. Sinus se označava sa sinθ , a kosinus sa cosθ pri čemu je θ naziv kuta.
Tangens (tg, tan) kuta je omjer sinusa i kosinusa:
tan
θ
=
sin
θ
cos
θ
.
{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}.}
S druge strane, imamo i recipročne funkcije pri čemu je kosinusu recipročan sekans (sec), sinusu kosekans (csc, cosec), a tangensu kotangens (ctg, cot):
sec
θ
=
1
cos
θ
,
csc
θ
=
1
sin
θ
,
cot
θ
=
1
tan
θ
=
cos
θ
sin
θ
.
{\displaystyle \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }},\quad \cot \theta ={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}.}
Inverzne funkcije Inverzne trigonometrijske funkcije ili arkus funkcije su inverzne funkcije trigonometrijskim funkcijama. Prema tome imamo, arkus sinus (arcsin, asin) je inverzna funkcija sinusnoj funkciji , pri čemu vrijedi da je
sin
(
arcsin
x
)
=
x
{\displaystyle \sin(\arcsin x)=x\!}
i
arcsin
(
sin
θ
)
=
θ
za
−
π
/
2
≤
θ
≤
π
/
2.
{\displaystyle \arcsin(\sin \theta )=\theta \quad {\text{za }}-\pi /2\leq \theta \leq \pi /2.}
U slijedećoj tablici su prikazane i druge komplementarne inverzne funkcije i kratice:
Trigonometrijska funkcija
Sinus
Kosinus
Tangens
Sekans
Kosekans
Kotangens
Kratica
sin
θ
{\displaystyle \operatorname {sin} \theta }
cos
θ
{\displaystyle \operatorname {cos} \theta }
tan
θ
{\displaystyle \operatorname {tan} \theta }
sec
θ
{\displaystyle \operatorname {sec} \theta }
csc
θ
{\displaystyle \operatorname {csc} \theta }
cot
θ
{\displaystyle \operatorname {cot} \theta }
Inverzna trigonometrijska funkcija
Arkus sinus
Arkus kosinus
Arkus tangens
Arkus sekans
Arkus kosekans
Arkus kotangens
Kratica
arcsin
θ
{\displaystyle \operatorname {arcsin} \theta }
arccos
θ
{\displaystyle \operatorname {arccos} \theta }
arctan
θ
{\displaystyle \operatorname {arctan} \theta }
arcsec
θ
{\displaystyle \operatorname {arcsec} \theta }
arccsc
θ
{\displaystyle \operatorname {arccsc} \theta }
arccot
θ
{\displaystyle \operatorname {arccot} \theta }
Pitagorin trigonometrijski identitet Ostale funkcije korištene u prošlosti Simetrija, pomak i periodičnost Zbroj i razlika kutova
Ovi trigonometrijski identiteti se nazivaju adicijske formule.
Otkrio ih je prezijski matematičar Abū al-Wafā' Būzjānī u 10. stoljeću.
Eulerova formula može pomoći pri dokazivanju ovih identiteta.
Sinus
sin
(
α
±
β
)
=
sin
α
cos
β
±
cos
α
sin
β
{\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \!}
[7] [8]
Kosinus
cos
(
α
±
β
)
=
cos
α
cos
β
∓
sin
α
sin
β
{\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \,}
[8] [9]
Tangens
tan
(
α
±
β
)
=
tan
α
±
tan
β
1
∓
tan
α
tan
β
{\displaystyle \tan(\alpha \pm \beta )={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}
[8] [10]
Arkus sinus
arcsin
α
±
arcsin
β
=
arcsin
(
α
1
−
β
2
±
β
1
−
α
2
)
{\displaystyle \arcsin \alpha \pm \arcsin \beta =\arcsin(\alpha {\sqrt {1-\beta ^{2}}}\pm \beta {\sqrt {1-\alpha ^{2}}})}
[11]
Arkus kosinus
arccos
α
±
arccos
β
=
arccos
(
α
β
∓
(
1
−
α
2
)
(
1
−
β
2
)
)
{\displaystyle \arccos \alpha \pm \arccos \beta =\arccos(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}})}
[12]
Arkus tangens
arctan
α
±
arctan
β
=
arctan
(
α
±
β
1
∓
α
β
)
{\displaystyle \arctan \alpha \pm \arctan \beta =\arctan \left({\frac {\alpha \pm \beta }{1\mp \alpha \beta }}\right)}
[13]
Matrični oblik Trigonometrijske formule zbroja i razlike za sinus i kosinus mogu biti zapisani u obliku matrice .
(
cos
ϕ
−
sin
ϕ
sin
ϕ
cos
ϕ
)
(
cos
θ
−
sin
θ
sin
θ
cos
θ
)
=
(
cos
ϕ
cos
θ
−
sin
ϕ
sin
θ
−
cos
ϕ
sin
θ
−
sin
ϕ
cos
θ
sin
ϕ
cos
θ
+
cos
ϕ
sin
θ
−
sin
ϕ
sin
θ
+
cos
ϕ
cos
θ
)
=
(
cos
(
θ
+
ϕ
)
−
sin
(
θ
+
ϕ
)
sin
(
θ
+
ϕ
)
cos
(
θ
+
ϕ
)
)
{\displaystyle {\begin{aligned}&{}\quad \left({\begin{array}{rr}\cos \phi &-\sin \phi \\\sin \phi &\cos \phi \end{array}}\right)\left({\begin{array}{rr}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos \phi \cos \theta -\sin \phi \sin \theta &-\cos \phi \sin \theta -\sin \phi \cos \theta \\\sin \phi \cos \theta +\cos \phi \sin \theta &-\sin \phi \sin \theta +\cos \phi \cos \theta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos(\theta +\phi )&-\sin(\theta +\phi )\\\sin(\theta +\phi )&\cos(\theta +\phi )\end{array}}\right)\end{aligned}}}
Sinus i kosinus zbroja beskonačno mnogo veličina
sin
(
∑
i
=
1
∞
θ
i
)
=
∑
odd
k
≥
1
(
−
1
)
(
k
−
1
)
/
2
∑
A
⊆
{
1
,
2
,
3
,
…
}
|
A
|
=
k
(
∏
i
∈
A
sin
θ
i
∏
i
∉
A
cos
θ
i
)
{\displaystyle \sin \left(\sum _{i=1}^{\infty }\theta _{i}\right)=\sum _{{\text{odd}}\ k\geq 1}(-1)^{(k-1)/2}\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)}
cos
(
∑
i
=
1
∞
θ
i
)
=
∑
even
k
≥
0
(
−
1
)
k
/
2
∑
A
⊆
{
1
,
2
,
3
,
…
}
|
A
|
=
k
(
∏
i
∈
A
sin
θ
i
∏
i
∉
A
cos
θ
i
)
{\displaystyle \cos \left(\sum _{i=1}^{\infty }\theta _{i}\right)=\sum _{{\text{even}}\ k\geq 0}~(-1)^{k/2}~~\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)}
Tangens zbroja konačno mnogo veličina Neka je
e
k
{\displaystyle e_{k}\,}
k ∈ {0, ..., n }) k -ti stupanj osnovnog simetričnog polinoma pri čemu je
x
i
=
tan
θ
i
{\displaystyle x_{i}=\tan \theta _{i}\,}
za i ∈ {0, ..., n } pa slijedi
e
0
=
1
e
1
=
∑
1
≤
i
≤
n
x
i
=
∑
1
≤
i
≤
n
tan
θ
i
e
2
=
∑
1
≤
i
<
j
≤
n
x
i
x
j
=
∑
1
≤
i
<
j
≤
n
tan
θ
i
tan
θ
j
e
3
=
∑
1
≤
i
<
j
<
k
≤
n
x
i
x
j
x
k
=
∑
1
≤
i
<
j
<
k
≤
n
tan
θ
i
tan
θ
j
tan
θ
k
⋮
⋮
{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{1\leq i\leq n}x_{i}&&=\sum _{1\leq i\leq n}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{1\leq i<j\leq n}x_{i}x_{j}&&=\sum _{1\leq i<j\leq n}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{1\leq i<j<k\leq n}x_{i}x_{j}x_{k}&&=\sum _{1\leq i<j<k\leq n}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&{}\ \ \vdots &&{}\ \ \vdots \end{aligned}}}
Tada vrijedi da je
tan
(
θ
1
+
⋯
+
θ
n
)
=
e
1
−
e
3
+
e
5
−
⋯
e
0
−
e
2
+
e
4
−
⋯
,
{\displaystyle \tan(\theta _{1}+\cdots +\theta _{n})={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots }},\!}
u ovisnosti o broju n .
Na primjer:
tan
(
θ
1
+
θ
2
)
=
e
1
e
0
−
e
2
=
x
1
+
x
2
1
−
x
1
x
2
=
tan
θ
1
+
tan
θ
2
1
−
tan
θ
1
tan
θ
2
,
tan
(
θ
1
+
θ
2
+
θ
3
)
=
e
1
−
e
3
e
0
−
e
2
=
(
x
1
+
x
2
+
x
3
)
−
(
x
1
x
2
x
3
)
1
−
(
x
1
x
2
+
x
1
x
3
+
x
2
x
3
)
,
tan
(
θ
1
+
θ
2
+
θ
3
+
θ
4
)
=
e
1
−
e
3
e
0
−
e
2
+
e
4
=
(
x
1
+
x
2
+
x
3
+
x
4
)
−
(
x
1
x
2
x
3
+
x
1
x
2
x
4
+
x
1
x
3
x
4
+
x
2
x
3
x
4
)
1
−
(
x
1
x
2
+
x
1
x
3
+
x
1
x
4
+
x
2
x
3
+
x
2
x
4
+
x
3
x
4
)
+
(
x
1
x
2
x
3
x
4
)
,
{\displaystyle {\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\tan \theta _{1}+\tan \theta _{2}}{1\ -\ \tan \theta _{1}\tan \theta _{2}}},\\\\\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}},\\\\\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\\\&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}},\end{aligned}}}
i tako dalje. Naveden identitet se može dokazati matematičkom indukcijom .[14]
Sekans i kosekans zbroja konačno mnogo veličina
sec
(
θ
1
+
⋯
+
θ
n
)
=
sec
θ
1
⋯
sec
θ
n
e
0
−
e
2
+
e
4
−
⋯
csc
(
θ
1
+
⋯
+
θ
n
)
=
sec
θ
1
⋯
sec
θ
n
e
1
−
e
3
+
e
5
−
⋯
{\displaystyle {\begin{aligned}\sec(\theta _{1}+\cdots +\theta _{n})&={\frac {\sec \theta _{1}\cdots \sec \theta _{n}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]\csc(\theta _{1}+\cdots +\theta _{n})&={\frac {\sec \theta _{1}\cdots \sec \theta _{n}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}
gdje je
e
k
{\displaystyle e_{k}\,}
k -ti stupanj osnovnog simetričnog polinoma za n varijabla x i θ i i = 1, ..., n , a broj veličina u nazivniku ovisi o n .
Na primjer,
sec
(
α
+
β
+
γ
)
=
sec
α
sec
β
sec
γ
1
−
tan
α
tan
β
−
tan
α
tan
γ
−
tan
β
tan
γ
csc
(
α
+
β
+
γ
)
=
sec
α
sec
β
sec
γ
tan
α
+
tan
β
+
tan
γ
−
tan
α
tan
β
tan
γ
{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}\end{aligned}}}
Identiteti za višestruke kutove
Tn je n -ti Chebyshevljev polinom
cos
n
θ
=
T
n
(
cos
θ
)
{\displaystyle \cos n\theta =T_{n}(\cos \theta )\,}
[15]
S n n -ti polinom širine
sin
2
n
θ
=
S
n
(
sin
2
θ
)
{\displaystyle \sin ^{2}n\theta =S_{n}(\sin ^{2}\theta )\,}
De Moivreova formula ,
i
{\displaystyle i}
imaginarna jedinica
cos
n
θ
+
i
sin
n
θ
=
(
cos
(
θ
)
+
i
sin
(
θ
)
)
n
{\displaystyle \cos n\theta +i\sin n\theta =(\cos(\theta )+i\sin(\theta ))^{n}\,}
[16]
Trigonomterijski identiteti dvostrukih, trostrukih i polovičnih kutova Sinus, kosinus i tangens višestrukih kutova
sin
n
θ
=
∑
k
=
0
n
(
n
k
)
cos
k
θ
sin
n
−
k
θ
sin
(
1
2
(
n
−
k
)
π
)
{\displaystyle \sin n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\sin \left({\frac {1}{2}}(n-k)\pi \right)}
cos
n
θ
=
∑
k
=
0
n
(
n
k
)
cos
k
θ
sin
n
−
k
θ
cos
(
1
2
(
n
−
k
)
π
)
{\displaystyle \cos n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\cos \left({\frac {1}{2}}(n-k)\pi \right)}
tan
(
n
+
1
)
θ
=
tan
n
θ
+
tan
θ
1
−
tan
n
θ
tan
θ
.
{\displaystyle \tan \,(n{+}1)\theta ={\frac {\tan n\theta +\tan \theta }{1-\tan n\theta \,\tan \theta }}.}
cot
(
n
+
1
)
θ
=
cot
n
θ
cot
θ
−
1
cot
n
θ
+
cot
θ
.
{\displaystyle \cot \,(n{+}1)\theta ={\frac {\cot n\theta \,\cot \theta -1}{\cot n\theta +\cot \theta }}.}
Chebyshevljeva metoda Chebyshevljeva metoda je rekurzivni algoritam za nalaženje formula n -tih višestrukih kutova poznavajući (n − 1)-te i (n − 2)-te formule.[22]
cos
n
x
=
2
⋅
cos
x
⋅
cos
(
n
−
1
)
x
−
cos
(
n
−
2
)
x
{\displaystyle \cos nx=2\cdot \cos x\cdot \cos(n-1)x-\cos(n-2)x\,}
sin
n
x
=
2
⋅
cos
x
⋅
sin
(
n
−
1
)
x
−
sin
(
n
−
2
)
x
{\displaystyle \sin nx=2\cdot \cos x\cdot \sin(n-1)x-\sin(n-2)x\,}
tan
n
x
=
H
+
K
tan
x
K
−
H
tan
x
{\displaystyle \tan nx={\frac {H+K\tan x}{K-H\tan x}}\,}
gdje je H /K = tan(n − 1)x .
Tangens prosjeka
tan
(
α
+
β
2
)
=
sin
α
+
sin
β
cos
α
+
cos
β
=
−
cos
α
−
cos
β
sin
α
−
sin
β
{\displaystyle \tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }}}
Ako su α ili β jednaki 0 tada dobivamo formulu za tangens polovičnog kuta.
Vièteov beskonačni produkt
cos
(
θ
2
)
⋅
cos
(
θ
4
)
⋅
cos
(
θ
8
)
⋯
=
∏
n
=
1
∞
cos
(
θ
2
n
)
=
sin
(
θ
)
θ
=
sinc
θ
.
{\displaystyle \cos \left({\theta \over 2}\right)\cdot \cos \left({\theta \over 4}\right)\cdot \cos \left({\theta \over 8}\right)\cdots =\prod _{n=1}^{\infty }\cos \left({\theta \over 2^{n}}\right)={\sin(\theta ) \over \theta }=\operatorname {sinc} \,\theta .}
Identiteti potenciranih trigonometrijskih funkcija Formule pretvorbi umnoška u zbroj i zbroja u umnožak
Umnožak u zbroj[23]
cos
θ
cos
φ
=
cos
(
θ
−
φ
)
+
cos
(
θ
+
φ
)
2
{\displaystyle \cos \theta \cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2}}
sin
θ
sin
φ
=
cos
(
θ
−
φ
)
−
cos
(
θ
+
φ
)
2
{\displaystyle \sin \theta \sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2}}
sin
θ
cos
φ
=
sin
(
θ
+
φ
)
+
sin
(
θ
−
φ
)
2
{\displaystyle \sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}}
cos
θ
sin
φ
=
sin
(
θ
+
φ
)
−
sin
(
θ
−
φ
)
2
{\displaystyle \cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}}
Zbroj u umnožak[24]
sin
θ
±
sin
φ
=
2
sin
(
θ
±
φ
2
)
cos
(
θ
∓
φ
2
)
{\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}
cos
θ
+
cos
φ
=
2
cos
(
θ
+
φ
2
)
cos
(
θ
−
φ
2
)
{\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}
cos
θ
−
cos
φ
=
−
2
sin
(
θ
+
φ
2
)
sin
(
θ
−
φ
2
)
{\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\theta +\varphi \over 2}\right)\sin \left({\theta -\varphi \over 2}\right)}
Drugi povezani identiteti Ako su x , y i z bilo kojeg trokuta, tada vrijedi
ako je zbroj
x
+
y
+
z
=
π
=
polukrug,
{\displaystyle {\text{ako je zbroj }}x+y+z=\pi ={\text{polukrug,}}\,}
onda je
tan
(
x
)
+
tan
(
y
)
+
tan
(
z
)
=
tan
(
x
)
tan
(
y
)
tan
(
z
)
.
{\displaystyle {\text{onda je }}\tan(x)+\tan(y)+\tan(z)=\tan(x)\tan(y)\tan(z).\,}
odnosno
ako je zbroj
x
+
y
+
z
=
π
=
polukrug,
{\displaystyle {\text{ako je zbroj }}x+y+z=\pi ={\text{polukrug,}}\,}
onda je
sin
(
2
x
)
+
sin
(
2
y
)
+
sin
(
2
z
)
=
4
sin
(
x
)
sin
(
y
)
sin
(
z
)
.
{\displaystyle {\text{onda je }}\sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\sin(y)\sin(z).\,}
Hermiteov kotangensov identitet Charles Hermite je pokazao da vrijedi određena jednakost[25] a 1 , ..., a n kompleksni brojevi . Neka je
A
n
,
k
=
∏
1
≤
j
≤
n
j
≠
k
cot
(
a
k
−
a
j
)
{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}
te u slučaju kada je A 1,1 , dobiva se prazan produkt , koji je jednak 1. Općenito se dobiva slijedeća vrijednost:
cot
(
z
−
a
1
)
⋯
cot
(
z
−
a
n
)
=
cos
n
π
2
+
∑
k
=
1
n
A
n
,
k
cot
(
z
−
a
k
)
.
{\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}
U najjednostavnijem slučaju za n = 2 vrijedi:
cot
(
z
−
a
1
)
cot
(
z
−
a
2
)
=
−
1
+
cot
(
a
1
−
a
2
)
cot
(
z
−
a
1
)
+
cot
(
a
2
−
a
1
)
cot
(
z
−
a
2
)
.
{\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}
Ptolemejev teorem Ove jednakosti predstavljaju trigonometrijski oblik ptolomejevog teorema .
Ako su
w
+
x
+
y
+
z
=
π
=
polukrug,
{\displaystyle {\text{Ako su }}w+x+y+z=\pi ={\text{polukrug,}}\,}
tada vrijedi
sin
(
w
+
x
)
sin
(
x
+
y
)
=
sin
(
x
+
y
)
sin
(
y
+
z
)
=
sin
(
y
+
z
)
sin
(
z
+
w
)
=
sin
(
z
+
w
)
sin
(
w
+
x
)
=
sin
(
w
)
sin
(
y
)
+
sin
(
x
)
sin
(
z
)
.
{\displaystyle {\begin{aligned}{\text{tada vrijedi }}&\sin(w+x)\sin(x+y)\\&{}=\sin(x+y)\sin(y+z)\\&{}=\sin(y+z)\sin(z+w)\\&{}=\sin(z+w)\sin(w+x)=\sin(w)\sin(y)+\sin(x)\sin(z).\end{aligned}}}
Linearne kombinacije
Bilo koja linearna kombinacija sinusnih valova istih perioda ili frekvencija s različitim faznim pomacima je također sinusni val sa istom periodom ili frekvencijom s različitim faznim pomakom. Kod nenulte linearne kombinacije sinusnog i kosinusnog vala
[26]
a
sin
x
+
b
cos
x
=
a
2
+
b
2
⋅
sin
(
x
+
φ
)
{\displaystyle a\sin x+b\cos x={\sqrt {a^{2}+b^{2}}}\cdot \sin(x+\varphi )\,}
gdje je
φ
=
{
arcsin
(
b
a
2
+
b
2
)
ako je
a
≥
0
,
π
−
arcsin
(
b
a
2
+
b
2
)
ako je
a
<
0
,
{\displaystyle \varphi ={\begin{cases}\arcsin \left({\frac {b}{\sqrt {a^{2}+b^{2}}}}\right)&{\text{ako je }}a\geq 0,\\\pi -\arcsin \left({\frac {b}{\sqrt {a^{2}+b^{2}}}}\right)&{\text{ako je }}a<0,\end{cases}}}
što je ekvivalentno s
φ
=
arctan
(
b
a
)
+
{
0
ako je
a
≥
0
,
π
ako je
a
<
0
,
{\displaystyle \varphi =\arctan \left({\frac {b}{a}}\right)+{\begin{cases}0&{\text{ako je }}a\geq 0,\\\pi &{\text{ako je }}a<0,\end{cases}}}
ili čak s
φ
=
sgn
(
b
)
cos
−
1
(
a
a
2
+
b
2
)
{\displaystyle \varphi ={\text{sgn}}(b)\cos ^{-1}\left({\tfrac {a}{\sqrt {a^{2}+b^{2}}}}\right)}
Općenito za proizvoljan fazni pomak vrijedi
a
sin
x
+
b
sin
(
x
+
α
)
=
c
sin
(
x
+
β
)
{\displaystyle a\sin x+b\sin(x+\alpha )=c\sin(x+\beta )\,}
gdje je
c
=
a
2
+
b
2
+
2
a
b
cos
α
,
{\displaystyle c={\sqrt {a^{2}+b^{2}+2ab\cos \alpha }},\,}
i
β
=
arctan
(
b
sin
α
a
+
b
cos
α
)
+
{
0
ako je
a
+
b
cos
α
≥
0
,
π
ako je
a
+
b
cos
α
<
0.
{\displaystyle \beta =\arctan \left({\frac {b\sin \alpha }{a+b\cos \alpha }}\right)+{\begin{cases}0&{\text{ako je }}a+b\cos \alpha \geq 0,\\\pi &{\text{ako je }}a+b\cos \alpha <0.\end{cases}}}
Lagrangeovi trigonometrijski identiteti
Ovi identiteti su ime dobili po Josephu Louisu Lagrangeu .[27] [28]
∑
n
=
1
N
sin
n
θ
=
1
2
cot
θ
−
cos
(
N
+
1
2
)
θ
2
sin
1
2
θ
∑
n
=
1
N
cos
n
θ
=
−
1
2
+
sin
(
N
+
1
2
)
θ
2
sin
1
2
θ
{\displaystyle {\begin{aligned}\sum _{n=1}^{N}\sin n\theta &={\frac {1}{2}}\cot \theta -{\frac {\cos(N+{\frac {1}{2}})\theta }{2\sin {\frac {1}{2}}\theta }}\\\sum _{n=1}^{N}\cos n\theta &=-{\frac {1}{2}}+{\frac {\sin(N+{\frac {1}{2}})\theta }{2\sin {\frac {1}{2}}\theta }}\end{aligned}}}
S njima je povezana funkcija koja se naziva Dirichletova jezgra .
1
+
2
cos
(
x
)
+
2
cos
(
2
x
)
+
2
cos
(
3
x
)
+
⋯
+
2
cos
(
n
x
)
=
sin
(
(
n
+
1
2
)
x
)
sin
(
x
/
2
)
.
{\displaystyle 1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}.}
Ostali oblici zbrojeva trigonometrijskih funkcija
Zbroj sinusa i kosinusa sa varijablama u aritmetičkom nizu
[29]
sin
φ
+
sin
(
φ
+
α
)
+
sin
(
φ
+
2
α
)
+
⋯
⋯
+
sin
(
φ
+
n
α
)
=
sin
(
(
n
+
1
)
α
2
)
⋅
sin
(
φ
+
n
α
2
)
sin
α
2
.
cos
φ
+
cos
(
φ
+
α
)
+
cos
(
φ
+
2
α
)
+
⋯
⋯
+
cos
(
φ
+
n
α
)
=
sin
(
(
n
+
1
)
α
2
)
⋅
cos
(
φ
+
n
α
2
)
sin
α
2
.
{\displaystyle {\begin{aligned}&\sin {\varphi }+\sin {(\varphi +\alpha )}+\sin {(\varphi +2\alpha )}+\cdots {}\\[8pt]&{}\qquad \qquad \cdots +\sin {(\varphi +n\alpha )}={\frac {\sin {\left({\frac {(n+1)\alpha }{2}}\right)}\cdot \sin {(\varphi +{\frac {n\alpha }{2}})}}{\sin {\frac {\alpha }{2}}}}.\\[10pt]&\cos {\varphi }+\cos {(\varphi +\alpha )}+\cos {(\varphi +2\alpha )}+\cdots {}\\[8pt]&{}\qquad \qquad \cdots +\cos {(\varphi +n\alpha )}={\frac {\sin {\left({\frac {(n+1)\alpha }{2}}\right)}\cdot \cos {(\varphi +{\frac {n\alpha }{2}})}}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}}
Za bilo koji a i b vrijedi:
a
cos
(
x
)
+
b
sin
(
x
)
=
a
2
+
b
2
cos
(
x
−
atan2
(
b
,
a
)
)
{\displaystyle a\cos(x)+b\sin(x)={\sqrt {a^{2}+b^{2}}}\cos(x-\operatorname {atan2} \,(b,a))\;}
gdje je atan2(y , x ) poopćenje funkcije arctan(y /x ) koja pokriva cijeli kružni opseg.
Koristeći Gudermannovu funkciju koja povezuje cirkularne i hiperbolne trigonometrijske funkcije bez korištenja kompleksnih brojeva može se iskoristiti slijedeći izraz:
tan
(
x
)
+
sec
(
x
)
=
tan
(
x
2
+
π
4
)
.
{\displaystyle \tan(x)+\sec(x)=\tan \left({x \over 2}+{\pi \over 4}\right).}
Ako su x , y i z ako su kutovi bilo kojeg trokuta odnosno x + y + z = π, tada je
cot
(
x
)
cot
(
y
)
+
cot
(
y
)
cot
(
z
)
+
cot
(
z
)
cot
(
x
)
=
1.
{\displaystyle \cot(x)\cot(y)+\cot(y)\cot(z)+\cot(z)\cot(x)=1.\,}
Određene linearne frakcionalne transformacije
Ako je ƒ (x ) dan linearnom frakcionalnom transformacijom
f
(
x
)
=
(
cos
α
)
x
−
sin
α
(
sin
α
)
x
+
cos
α
,
{\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}
i slično tome
g
(
x
)
=
(
cos
β
)
x
−
sin
β
(
cos
β
)
x
+
sin
β
,
{\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\cos \beta )x+\sin \beta }},}
tada vrijedi
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
=
(
cos
(
α
+
β
)
)
x
−
sin
(
α
+
β
)
(
sin
(
α
+
β
)
)
x
+
cos
(
α
+
β
)
.
{\displaystyle f(g(x))=g(f(x))={\frac {(\cos(\alpha +\beta ))x-\sin(\alpha +\beta )}{(\sin(\alpha +\beta ))x+\cos(\alpha +\beta )}}.}
Kraće rečeno, ako je za sve α funkcija ƒ α ƒ tada vrijedi da je
f
α
∘
f
β
=
f
α
+
β
.
{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.\,}
Identiteti s inverznim trigonometrijskim funkcijama
arcsin
(
x
)
+
arccos
(
x
)
=
π
/
2
{\displaystyle \arcsin(x)+\arccos(x)=\pi /2\;}
arctan
(
x
)
+
arccot
(
x
)
=
π
/
2.
{\displaystyle \arctan(x)+\operatorname {arccot}(x)=\pi /2.\;}
arctan
(
x
)
+
arctan
(
1
/
x
)
=
{
π
/
2
,
ako je
x
>
0
−
π
/
2
,
ako je
x
<
0
{\displaystyle \arctan(x)+\arctan(1/x)=\left\{{\begin{matrix}\pi /2,&{\mbox{ako je }}x>0\\-\pi /2,&{\mbox{ako je }}x<0\end{matrix}}\right.}
Kompozicija trigonometrijskih i inverznih trigonometrijskih funkcija
sin
[
arccos
(
x
)
]
=
1
−
x
2
{\displaystyle \sin[\arccos(x)]={\sqrt {1-x^{2}}}\,}
tan
[
arcsin
(
x
)
]
=
x
1
−
x
2
{\displaystyle \tan[\arcsin(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
sin
[
arctan
(
x
)
]
=
x
1
+
x
2
{\displaystyle \sin[\arctan(x)]={\frac {x}{\sqrt {1+x^{2}}}}}
tan
[
arccos
(
x
)
]
=
1
−
x
2
x
{\displaystyle \tan[\arccos(x)]={\frac {\sqrt {1-x^{2}}}{x}}}
cos
[
arctan
(
x
)
]
=
1
1
+
x
2
{\displaystyle \cos[\arctan(x)]={\frac {1}{\sqrt {1+x^{2}}}}}
cot
[
arcsin
(
x
)
]
=
1
−
x
2
x
{\displaystyle \cot[\arcsin(x)]={\frac {\sqrt {1-x^{2}}}{x}}}
cos
[
arcsin
(
x
)
]
=
1
−
x
2
{\displaystyle \cos[\arcsin(x)]={\sqrt {1-x^{2}}}\,}
cot
[
arccos
(
x
)
]
=
x
1
−
x
2
{\displaystyle \cot[\arccos(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
Povezanost sa kompleksnom eksponencijalnom funkcijom Povezanost s beskonačnim produktima Identiteti bez varijabli Infinitezimalni račun
Derivacije Koristeći infinitezimalni račun , kutovi pri računanju moraju biti u radijanima. Derivacije trigonometrijskih funkcija mogu se odrediti pomoću dva limesa :
lim
x
→
0
sin
x
x
=
1
,
{\displaystyle \lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1,}
lim
x
→
0
1
−
cos
x
x
=
0
,
{\displaystyle \lim _{x\rightarrow 0}{\frac {1-\cos x}{x}}=0,}
Deriviranjem trigonometrijskih funkcija dobivaju se slijedeći identiteti i pravila:[36] [37] [38]
d
d
x
sin
x
=
cos
x
,
d
d
x
arcsin
x
=
1
1
−
x
2
d
d
x
cos
x
=
−
sin
x
,
d
d
x
arccos
x
=
−
1
1
−
x
2
d
d
x
tan
x
=
sec
2
x
,
d
d
x
arctan
x
=
1
1
+
x
2
d
d
x
cot
x
=
−
csc
2
x
,
d
d
x
arccot
x
=
−
1
1
+
x
2
d
d
x
sec
x
=
tan
x
sec
x
,
d
d
x
arcsec
x
=
1
|
x
|
x
2
−
1
d
d
x
csc
x
=
−
csc
x
cot
x
,
d
d
x
arccsc
x
=
−
1
|
x
|
x
2
−
1
{\displaystyle {\begin{aligned}{d \over dx}\sin x&=\cos x,&{d \over dx}\arcsin x&={1 \over {\sqrt {1-x^{2}}}}\\\\{d \over dx}\cos x&=-\sin x,&{d \over dx}\arccos x&={-1 \over {\sqrt {1-x^{2}}}}\\\\{d \over dx}\tan x&=\sec ^{2}x,&{d \over dx}\arctan x&={1 \over 1+x^{2}}\\\\{d \over dx}\cot x&=-\csc ^{2}x,&{d \over dx}\operatorname {arccot} x&={-1 \over 1+x^{2}}\\\\{d \over dx}\sec x&=\tan x\sec x,&{d \over dx}\operatorname {arcsec} x&={1 \over |x|{\sqrt {x^{2}-1}}}\\\\{d \over dx}\csc x&=-\csc x\cot x,&{d \over dx}\operatorname {arccsc} x&={-1 \over |x|{\sqrt {x^{2}-1}}}\end{aligned}}}
Integrali
∫
d
u
a
2
−
u
2
=
sin
−
1
(
u
a
)
+
C
{\displaystyle \int {\frac {du}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}\left({\frac {u}{a}}\right)+C}
∫
d
u
a
2
+
u
2
=
1
a
tan
−
1
(
u
a
)
+
C
{\displaystyle \int {\frac {du}{a^{2}+u^{2}}}={\frac {1}{a}}\tan ^{-1}\left({\frac {u}{a}}\right)+C}
∫
d
u
u
u
2
−
a
2
=
1
a
sec
−
1
|
u
a
|
+
C
{\displaystyle \int {\frac {du}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}\left|{\frac {u}{a}}\right|+C}
Eksponencijalne definicije trigonometrijskih funkcija
Funkcija
Inverzna funkcija[39]
sin
θ
=
e
i
θ
−
e
−
i
θ
2
i
{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}
arcsin
x
=
−
i
ln
(
i
x
+
1
−
x
2
)
{\displaystyle \arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}
cos
θ
=
e
i
θ
+
e
−
i
θ
2
{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}
arccos
x
=
−
i
ln
(
x
+
x
2
−
1
)
{\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1}}\right)\,}
tan
θ
=
e
i
θ
−
e
−
i
θ
i
(
e
i
θ
+
e
−
i
θ
)
{\displaystyle \tan \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}
arctan
x
=
i
2
ln
(
i
+
x
i
−
x
)
{\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\,}
csc
θ
=
2
i
e
i
θ
−
e
−
i
θ
{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\,}
arccsc
x
=
−
i
ln
(
i
x
+
1
−
1
x
2
)
{\displaystyle \operatorname {arccsc} x=-i\ln \left({\tfrac {i}{x}}+{\sqrt {1-{\tfrac {1}{x^{2}}}}}\right)\,}
sec
θ
=
2
e
i
θ
+
e
−
i
θ
{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\,}
arcsec
x
=
−
i
ln
(
1
x
+
1
−
i
x
2
)
{\displaystyle \operatorname {arcsec} x=-i\ln \left({\tfrac {1}{x}}+{\sqrt {1-{\tfrac {i}{x^{2}}}}}\right)\,}
cot
θ
=
i
(
e
i
θ
+
e
−
i
θ
)
e
i
θ
−
e
−
i
θ
{\displaystyle \cot \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}
arccot
x
=
i
2
ln
(
x
−
i
x
+
i
)
{\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\,}
cis
θ
=
e
i
θ
{\displaystyle \operatorname {cis} \,\theta =e^{i\theta }\,}
arccis
x
=
ln
x
i
=
−
i
ln
x
=
arg
x
{\displaystyle \operatorname {arccis} \,x={\frac {\ln x}{i}}=-i\ln x=\operatorname {arg} \,x\,}
Weierstrassova supstitucija
Ako je
t
=
tan
(
x
2
)
,
{\displaystyle t=\tan \left({\frac {x}{2}}\right),}
tada vrijedi [40]
sin
(
x
)
=
2
t
1
+
t
2
and
cos
(
x
)
=
1
−
t
2
1
+
t
2
and
e
i
x
=
1
+
i
t
1
−
i
t
{\displaystyle \sin(x)={\frac {2t}{1+t^{2}}}{\text{ and }}\cos(x)={\frac {1-t^{2}}{1+t^{2}}}{\text{ and }}e^{ix}={\frac {1+it}{1-it}}}
gdje je eix x ) + i sin(x ), što ponekad skraćeno pišemo kao cis(x ).
Vidi još Bilješke
↑ Abramowitz and Stegun, p. 73, 4.3.45
↑ Abramowitz and Stegun, p. 78, 4.3.147
↑ Abramowitz and Stegun, p. 72, 4.3.13–15
↑ The Elementary Identities ↑ Abramowitz and Stegun, p. 72, 4.3.9
↑ Abramowitz and Stegun, p. 72, 4.3.7–8
↑ Abramowitz and Stegun, p. 72, 4.3.16
↑ a b c Predložak:MathWorld ↑ Abramowitz and Stegun, p. 72, 4.3.17
↑ Abramowitz and Stegun, p. 72, 4.3.18
↑ Abramowitz and Stegun, p. 80, 4.4.42
↑ Abramowitz and Stegun, p. 80, 4.4.43
↑ Abramowitz and Stegun, p. 80, 4.4.36
↑ Bronstein, Manual. 1989. Simplification of Real Elementary Functions. Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation : 211 ↑ a b Predložak:MathWorld ↑ Abramowitz and Stegun, p. 74, 4.3.48
↑ Abramowitz and Stegun, p. 72, 4.3.24–26
↑ Predložak:MathWorld ↑ Abramowitz and Stegun, p. 72, 4.3.27–28
↑ Abramowitz and Stegun, p. 72, 4.3.20–22
↑ Predložak:MathWorld ↑ Ken Ward's Mathematics Pages, http://www.trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm
↑ Abramowitz and Stegun, p. 72, 4.3.31–33
↑ Abramowitz and Stegun, p. 72, 4.3.34–39
↑ Warren P. Johnson, "Trigonometric Identities à la Hermite", American Mathematical Monthly
↑ Proof at http://pages.pacificcoast.net/~cazelais/252/lc-trig.pdf
↑
Eddie Ortiz Muñiz. Veljača 1953. A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities. American Journal of Physics . 21 (2): 140
↑
Alan Jeffrey and Hui-hui Dai. 2008. Section 2.4.1.6. Handbook of Mathematical Formulas and Integrals 4th izdanje. Academic Press. ISBN 9780123742889
↑ Michael P. Knapp, Sines and Cosines of Angles in Arithmetic Progression
↑ Abramowitz and Stegun, p. 74, 4.3.47
↑ Abramowitz and Stegun, p. 71, 4.3.2
↑ Abramowitz and Stegun, p. 71, 4.3.1
↑ Abramowitz and Stegun, p. 75, 4.3.89–90
↑ Abramowitz and Stegun, p. 85, 4.5.68–69
↑ Weisstein, Eric W. , "Sine " from MathWorld ↑ Abramowitz and Stegun, p. 77, 4.3.105–110
↑ Abramowitz and Stegun, p. 82, 4.4.52–57
↑ Finney, Ross. 2003. Calculus : Graphical, Numerical, Algebraic . Prentice Hall. Glenview, Illinois. str. 159–161. ISBN 0-13-063131-0 ↑ Abramowitz and Stegun, p. 80, 4.4.26–31
↑ Abramowitz and Stegun, p. 72, 4.3.23
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