Trigonometrijske jednakosti 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 . Jednakosti se koriste za pojednostavljenje izraza koji uključuju trigonometrijske funkcije.
Sinusi i kosinusi u jediničnoj kružnici
Podrobniji članak o temi:
Kut
Nazivi 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 }
radijana = 400 gradi.
Sljedeća tablica prikazuje pretvorbu mjernih jedinica za određene veličine kutova:
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
uredi
Primarne trigonometrijske funkcije su sinus i kosinus kuta. Sinus se označava sa sinθ , a kosinus s cosθ pri čemu je θ naziv kuta.
Tangens (tg, tan) kuta je omjer sinusa i kosinusa:
tg
θ
=
sin
θ
cos
θ
.
{\displaystyle \operatorname {tg} \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
,
ctg
θ
=
1
tg
θ
=
cos
θ
sin
θ
.
{\displaystyle \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }},\quad \operatorname {ctg} \theta ={\frac {1}{\operatorname {tg} \theta }}={\frac {\cos \theta }{\sin \theta }}.}
Inverzne funkcije
uredi
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 sljedećoj 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 }
tg
θ
{\displaystyle \operatorname {tg} \theta }
sec
θ
{\displaystyle \operatorname {sec} \theta }
csc
θ
{\displaystyle \operatorname {csc} \theta }
ctg
θ
{\displaystyle \operatorname {ctg} \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 }
arctg
θ
{\displaystyle \operatorname {arctg} \theta }
arcsec
θ
{\displaystyle \operatorname {arcsec} \theta }
arccsc
θ
{\displaystyle \operatorname {arccsc} \theta }
arcctg
θ
{\displaystyle \operatorname {arcctg} \theta }
Pitagorina trigonometrijska jednakost
uredi
Ostale funkcije korištene u prošlosti
uredi
Simetrija, pomak i periodičnost
uredi
Zbroj i razlika kutova
uredi
Ove trigonometrijske jednakosti se nazivaju adicijske formule. Otkrio ih je prezijski matematičar Abū al-Wafā' Būzjānī u 10. stoljeću. Eulerova formula može pomoći pri dokazivanju ovih jednakosti.
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]
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]
Tangens
tg
(
α
±
β
)
=
tg
α
±
tg
β
1
∓
tg
α
tg
β
{\displaystyle \operatorname {tg} (\alpha \pm \beta )={\frac {\operatorname {tg} \alpha \pm \operatorname {tg} \beta }{1\mp \operatorname {tg} \alpha \operatorname {tg} \beta }}}
[9]
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}}})}
[10]
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})}})}
[11]
Arkus tangens
arctg
α
±
arctg
β
=
arctg
(
α
±
β
1
∓
α
β
)
{\displaystyle \operatorname {arctg} \alpha \pm \operatorname {arctg} \beta =\operatorname {arctg} \left({\frac {\alpha \pm \beta }{1\mp \alpha \beta }}\right)}
[12]
Matrični oblik
uredi
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
uredi
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
uredi
Neka je
e
k
{\displaystyle e_{k}\,}
(za k ∈ {0, ..., n }) k -ti stupanj osnovnog simetričnog polinoma pri čemu je
x
i
=
tg
θ
i
{\displaystyle x_{i}=\operatorname {tg} \theta _{i}\,}
za i ∈ {0, ..., n } pa slijedi
e
0
=
1
e
1
=
∑
1
≤
i
≤
n
x
i
=
∑
1
≤
i
≤
n
tg
θ
i
e
2
=
∑
1
≤
i
<
j
≤
n
x
i
x
j
=
∑
1
≤
i
<
j
≤
n
tg
θ
i
tg
θ
j
e
3
=
∑
1
≤
i
<
j
<
k
≤
n
x
i
x
j
x
k
=
∑
1
≤
i
<
j
<
k
≤
n
tg
θ
i
tg
θ
j
tg
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}\operatorname {tg} \theta _{i}\\[6pt]e_{2}&=\sum _{1\leq i<j\leq n}x_{i}x_{j}&&=\sum _{1\leq i<j\leq n}\operatorname {tg} \theta _{i}\operatorname {tg} \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}\operatorname {tg} \theta _{i}\operatorname {tg} \theta _{j}\operatorname {tg} \theta _{k}\\&{}\ \ \vdots &&{}\ \ \vdots \end{aligned}}}
Tada vrijedi da je
tg
(
θ
1
+
⋯
+
θ
n
)
=
e
1
−
e
3
+
e
5
−
⋯
e
0
−
e
2
+
e
4
−
⋯
,
{\displaystyle \operatorname {tg} (\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:
tg
(
θ
1
+
θ
2
)
=
e
1
e
0
−
e
2
=
x
1
+
x
2
1
−
x
1
x
2
=
tg
θ
1
+
tg
θ
2
1
−
tg
θ
1
tg
θ
2
,
tg
(
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
)
,
tg
(
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}\operatorname {tg} (\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\operatorname {tg} \theta _{1}+\operatorname {tg} \theta _{2}}{1\ -\ \operatorname {tg} \theta _{1}\operatorname {tg} \theta _{2}}},\\\\\operatorname {tg} (\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})}},\\\\\operatorname {tg} (\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. Navedena jednakost se može dokazati matematičkom indukcijom .[13]
Sekans i kosekans zbroja konačno mnogo veličina
uredi
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 = tan 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
−
tg
α
tg
β
−
tg
α
tg
γ
−
tg
β
tg
γ
csc
(
α
+
β
+
γ
)
=
sec
α
sec
β
sec
γ
tg
α
+
tg
β
+
tg
γ
−
tg
α
tg
β
tg
γ
{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\operatorname {tg} \alpha \operatorname {tg} \beta -\operatorname {tg} \alpha \operatorname {tg} \gamma -\operatorname {tg} \beta \operatorname {tg} \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\operatorname {tg} \alpha +\operatorname {tg} \beta +\operatorname {tg} \gamma -\operatorname {tg} \alpha \operatorname {tg} \beta \operatorname {tg} \gamma }}\end{aligned}}}
Jednakosti za višestruke kutove
uredi
Tn je n -ti Čebiševljev polinom
cos
n
θ
=
T
n
(
cos
θ
)
{\displaystyle \cos n\theta =T_{n}(\cos \theta )\,}
S n je 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}
je 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}\,}
[14]
Trigonomterijske jednakosti dvostrukih, trostrukih i polovičnih kutova
uredi
Sinus, kosinus i tangens višestrukih kutova
uredi
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)}
tg
(
n
+
1
)
θ
=
tg
n
θ
+
tg
θ
1
−
tg
n
θ
tg
θ
.
{\displaystyle \operatorname {tg} \,(n{+}1)\theta ={\frac {\operatorname {tg} n\theta +\operatorname {tg} \theta }{1-\operatorname {tg} n\theta \,\operatorname {tg} \theta }}.}
ctg
(
n
+
1
)
θ
=
ctg
n
θ
ctg
θ
−
1
ctg
n
θ
+
ctg
θ
.
{\displaystyle \operatorname {ctg} \,(n{+}1)\theta ={\frac {\operatorname {ctg} n\theta \,\operatorname {ctg} \theta -1}{\operatorname {ctg} n\theta +\operatorname {ctg} \theta }}.}
Čebiševljeva metoda
uredi
Čebiševljeva metoda je rekurzivni algoritam za nalaženje formula n -tih višestrukih kutova poznavajući (n − 1)-te i (n − 2)-te formule.[17]
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\,}
tg
n
x
=
H
+
K
tg
x
K
−
H
tg
x
{\displaystyle \operatorname {tg} nx={\frac {H+K\operatorname {tg} x}{K-H\operatorname {tg} x}}\,}
gdje je H /K = tan(n − 1)x .
Tangens prosjeka
uredi
tg
(
α
+
β
2
)
=
sin
α
+
sin
β
cos
α
+
cos
β
=
−
cos
α
−
cos
β
sin
α
−
sin
β
{\displaystyle \operatorname {tg} \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
uredi
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 .}
Jednakosti potenciranih trigonometrijskih funkcija
uredi
Formule pretvorbi umnoška u zbroj i zbroja u umnožak
uredi
Umnožak u zbroj[18]
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[19]
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)}
Druge povezane jednakosti
uredi
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
tg
(
x
)
+
tg
(
y
)
+
tg
(
z
)
=
tg
(
x
)
tg
(
y
)
tg
(
z
)
.
{\displaystyle {\text{onda je }}\operatorname {tg} (x)+\operatorname {tg} (y)+\operatorname {tg} (z)=\operatorname {tg} (x)\operatorname {tg} (y)\operatorname {tg} (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).\,}
Hermiteova kotangensova jednakost
uredi
Charles Hermite je pokazao da vrijedi određena jednakost[20] gdje su varijable a 1 , ..., a n kompleksni brojevi . Neka je
A
n
,
k
=
∏
1
≤
j
≤
n
j
≠
k
ctg
(
a
k
−
a
j
)
{\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\operatorname {ctg} (a_{k}-a_{j})}
te u slučaju kada je A 1,1 , dobiva se prazan produkt , koji je jednak 1. Općenito se dobiva sljedeća vrijednost:
ctg
(
z
−
a
1
)
⋯
ctg
(
z
−
a
n
)
=
cos
n
π
2
+
∑
k
=
1
n
A
n
,
k
ctg
(
z
−
a
k
)
.
{\displaystyle \operatorname {ctg} (z-a_{1})\cdots \operatorname {ctg} (z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\operatorname {ctg} (z-a_{k}).}
U najjednostavnijem slučaju za n = 2 vrijedi:
ctg
(
z
−
a
1
)
ctg
(
z
−
a
2
)
=
−
1
+
ctg
(
a
1
−
a
2
)
ctg
(
z
−
a
1
)
+
ctg
(
a
2
−
a
1
)
ctg
(
z
−
a
2
)
.
{\displaystyle \operatorname {ctg} (z-a_{1})\operatorname {ctg} (z-a_{2})=-1+\operatorname {ctg} (a_{1}-a_{2})\operatorname {ctg} (z-a_{1})+\operatorname {ctg} (a_{2}-a_{1})\operatorname {ctg} (z-a_{2}).}
Ptolemejev teorem
uredi
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
uredi
Bilo koja linearna kombinacija sinusnih valova istih perioda ili frekvencija s različitim faznim pomacima je također sinusni val s istom periodom ili frekvencijom s različitim faznim pomakom. Kod nenulte linearne kombinacije sinusnog i kosinusnog vala
,[21] se dobiva
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
=
arctg
(
b
a
)
+
{
0
ako je
a
≥
0
,
π
ako je
a
<
0
,
{\displaystyle \varphi =\operatorname {arctg} \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
=
arctg
(
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 =\operatorname {arctg} \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}}}
Lagrangeove trigonometrijske jednakosti
uredi
Ove jednakosti su ime dobili po Josephu Louisu Lagrangeu .[22] [23]
∑
n
=
1
N
sin
n
θ
=
1
2
ctg
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}}\operatorname {ctg} \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
uredi
Zbroj sinusa i kosinusa s varijablama u aritmetičkom nizu
[24] :
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 sljedeći izraz:
tg
(
x
)
+
sec
(
x
)
=
tg
(
x
2
+
π
4
)
.
{\displaystyle \operatorname {tg} (x)+\sec(x)=\operatorname {tg} \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
ctg
(
x
)
ctg
(
y
)
+
ctg
(
y
)
ctg
(
z
)
+
ctg
(
z
)
ctg
(
x
)
=
1.
{\displaystyle \operatorname {ctg} (x)\operatorname {ctg} (y)+\operatorname {ctg} (y)\operatorname {ctg} (z)+\operatorname {ctg} (z)\operatorname {ctg} (x)=1.\,}
Određene linearne frakcionalne transformacije
uredi
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 baš ta gore prikazana funkcija ƒ tada vrijedi da je
f
α
∘
f
β
=
f
α
+
β
.
{\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.\,}
Jednakosti inverznih trigonometrijskih funkcija
uredi
arcsin
(
x
)
+
arccos
(
x
)
=
π
/
2
{\displaystyle \arcsin(x)+\arccos(x)=\pi /2\;}
arctg
(
x
)
+
arcctg
(
x
)
=
π
/
2.
{\displaystyle \operatorname {arctg} (x)+\operatorname {arcctg} (x)=\pi /2.\;}
arctg
(
x
)
+
arctg
(
1
/
x
)
=
{
π
/
2
,
ako je
x
>
0
−
π
/
2
,
ako je
x
<
0
{\displaystyle \operatorname {arctg} (x)+\operatorname {arctg} (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
uredi
sin
[
arccos
(
x
)
]
=
1
−
x
2
{\displaystyle \sin[\arccos(x)]={\sqrt {1-x^{2}}}\,}
tg
[
arcsin
(
x
)
]
=
x
1
−
x
2
{\displaystyle \operatorname {tg} [\arcsin(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
sin
[
arctg
(
x
)
]
=
x
1
+
x
2
{\displaystyle \sin[\operatorname {arctg} (x)]={\frac {x}{\sqrt {1+x^{2}}}}}
tg
[
arccos
(
x
)
]
=
1
−
x
2
x
{\displaystyle \operatorname {tg} [\arccos(x)]={\frac {\sqrt {1-x^{2}}}{x}}}
cos
[
arctg
(
x
)
]
=
1
1
+
x
2
{\displaystyle \cos[\operatorname {arctg} (x)]={\frac {1}{\sqrt {1+x^{2}}}}}
ctg
[
arcsin
(
x
)
]
=
1
−
x
2
x
{\displaystyle \operatorname {ctg} [\arcsin(x)]={\frac {\sqrt {1-x^{2}}}{x}}}
cos
[
arcsin
(
x
)
]
=
1
−
x
2
{\displaystyle \cos[\arcsin(x)]={\sqrt {1-x^{2}}}\,}
ctg
[
arccos
(
x
)
]
=
x
1
−
x
2
{\displaystyle \operatorname {ctg} [\arccos(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
Povezanost s kompleksnom eksponencijalnom funkcijom
uredi
Povezanost s beskonačnim produktima
uredi
Jednakosti bez varijabli
uredi
Infinitezimalni račun
uredi
Derivacije
uredi
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 sljedeće jednakosti i pravila:[31] [32] [33]
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
tg
x
=
sec
2
x
,
d
d
x
arctg
x
=
1
1
+
x
2
d
d
x
ctg
x
=
−
csc
2
x
,
d
d
x
arcctg
x
=
−
1
1
+
x
2
d
d
x
sec
x
=
tg
x
sec
x
,
d
d
x
arcsec
x
=
1
|
x
|
x
2
−
1
d
d
x
csc
x
=
−
csc
x
ctg
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}\operatorname {tg} x&=\sec ^{2}x,&{d \over dx}\operatorname {arctg} x&={1 \over 1+x^{2}}\\\\{d \over dx}\operatorname {ctg} x&=-\csc ^{2}x,&{d \over dx}\operatorname {arcctg} x&={-1 \over 1+x^{2}}\\\\{d \over dx}\sec x&=\operatorname {tg} x\sec x,&{d \over dx}\operatorname {arcsec} x&={1 \over |x|{\sqrt {x^{2}-1}}}\\\\{d \over dx}\csc x&=-\csc x\operatorname {ctg} x,&{d \over dx}\operatorname {arccsc} x&={-1 \over |x|{\sqrt {x^{2}-1}}}\end{aligned}}}
∫
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
tg
−
1
(
u
a
)
+
C
{\displaystyle \int {\frac {du}{a^{2}+u^{2}}}={\frac {1}{a}}\operatorname {tg} ^{-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
uredi
Funkcija
Inverzna funkcija[34]
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)\,}
tg
θ
=
e
i
θ
−
e
−
i
θ
i
(
e
i
θ
+
e
−
i
θ
)
{\displaystyle \operatorname {tg} \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}
arctg
x
=
i
2
ln
(
i
+
x
i
−
x
)
{\displaystyle \operatorname {arctg} 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)\,}
ctg
θ
=
i
(
e
i
θ
+
e
−
i
θ
)
e
i
θ
−
e
−
i
θ
{\displaystyle \operatorname {ctg} \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}
arcctg
x
=
i
2
ln
(
x
−
i
x
+
i
)
{\displaystyle \operatorname {arcctg} 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
uredi
Ako je
t
=
tg
(
x
2
)
,
{\displaystyle t=\operatorname {tg} \left({\frac {x}{2}}\right),}
tada vrijedi[35]
sin
(
x
)
=
2
t
1
+
t
2
i
cos
(
x
)
=
1
−
t
2
1
+
t
2
i
e
i
x
=
1
+
i
t
1
−
i
t
{\displaystyle \sin(x)={\frac {2t}{1+t^{2}}}{\text{ i }}\cos(x)={\frac {1-t^{2}}{1+t^{2}}}{\text{ i }}e^{ix}={\frac {1+it}{1-it}}}
gdje je eix = cos(x ) + i sin(x ), što ponekad skraćeno pišemo kao cis(x ).
↑ 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 . Inačica izvorne stranice arhivirana 30. srpnja 2017. Pristupljeno 20. prosinca 2011.
↑ 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
↑ 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
↑ Abramowitz and Stegun, p. 74, 4.3.48
↑ Abramowitz and Stegun, p. 72, 4.3.24–26
↑ Abramowitz and Stegun, p. 72, 4.3.20–22
↑ Ken Ward's Mathematics Pages, http://www.trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm Arhivirana inačica izvorne stranice od 27. srpnja 2011. (Wayback Machine )
↑ 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 , volume 117, number 4, April 2010, pages 311–327
↑ Proof at http://pages.pacificcoast.net/~cazelais/252/lc-trig.pdf Arhivirana inačica izvorne stranice od 3. prosinca 2011. (Wayback Machine )
↑
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 Arhivirana inačica izvorne stranice od 19. srpnja 2011. (Wayback Machine )
↑ 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
Vanjske poveznice
uredi