Kosinus i sinus oko jediničnog kruga . U matematici trigonometrijski identiteti su ekvivalentni sa upotrebom trigonometrijskih funkcija i one važe za svaku vrednost promenljivih. Geometrijski, oni su identiteti koji uključuju određene funkcije jednog ili više uglova. Oni su posebni trigonometrijski identiteti, oni uključuju oba ugla dužine strana trougla. Samo neki su pomenuti u ovom članku.
Ovi identitei su korisni kada god imamo izraz koji uključuje trigonometrijske funkcije, a treba da bude pojednostavljen. Bitan zahtev je integracija ne-trigonometrijskih funkcija : uobičajena tehnika uključuje prvobitno primenu pravila supstitucije na trigonometrijskim funkcijama, i onda pojednostavljivanje rezultata integrala sa trigonometrijskim identitemia.
Ovaj članak koristi grčka slova kao što su alpha beta , gamma i theta da predstavi uglove. Nekoliko različitih jedinica su široko rasprostranjene, uklljučujući stepene, radijane i gradijane :
1 pun krug = 360 stepeni = 2
π
{\displaystyle \pi }
Sledeća tabela pokazuje verziju nekih od uobičajenih uglova:
Degrees
30°
60°
120°
150°
210°
240°
300°
330°
Radians
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}}\!}
Grads
33⅓ grad
66⅔ grad
133⅓ grad
166⅔ grad
233⅓ grad
266⅔ grad
333⅓ grad
366⅔ grad
Degrees
45°
90°
135°
180°
225°
270°
315°
360°
Radians
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 \!}
Grads
50 grad
100 grad
150 grad
200 grad
250 grad
300 grad
350 grad
400 grad
Osim ako je navedeno suprotno svi uglovi u ovom članku će biti u radijanima, osim uglova koji se završavaju simbolom stepena (°), koji su u stepenima.[ 1]
Primarne trigonometrijske funkcije su sinus i kosinus ugla. Oni su ponkad skraćene sin i cos.
Sinus ugla je definisan u kontekstu pravog trougla kao odnos dužina stranice koja je naspram ugla, podeljene dužinom najduže stranice trougla, hipotenuze.
Kosinus ugla je takođe definisan u kontekstu pravog trougla, kao odnos dužina stranica na kojoj leži ugao, podeljene dužinom najduže stranice trougla, hipotenuzom.
Tangens ugla je odnos sinusa i kosinusa:
tan
θ
=
sin
θ
cos
θ
.
{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}.}
Konačno, reciprocna funkcija sec, csc i ctg su recipročne sinusu kosinusu i tangensu:
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 trigonometrijske funkcije su delimično inverzne funkcije za trigonometrijske funkcije. Na primer, inverzna funkcija za sinus, poznata kao 'inverzni sinus' ili arcsin zadivoljava
sin
(
arcsin
x
)
=
x
for
|
x
|
≤
1
{\displaystyle \sin(\arcsin x)=x\quad {\text{for}}\quad |x|\leq 1}
i
arcsin
(
sin
x
)
=
x
for
|
x
|
≤
π
/
2.
{\displaystyle \arcsin(\sin x)=x\quad {\text{for}}\quad |x|\leq \pi /2.}
Ovaj članak koristi notacije ispod za inverzne trigonometrijske funkcije :
Function
sin
cos
tan
sec
csc
cot
Inverse
arcsin
arccos
arctan
arcsec
arccsc
arccot
Osnovna veza između sinusa i kosinusa su Pitagorini trigonometrijski identitet :
cos
2
θ
+
sin
2
θ
=
1
{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1\!}
gde cos2 θ znači (cos(θ ))2 i sin2 θ znači (sin(θ ))2 .
Ovo se može posmatrati kao verzija pitagorine teoreme, i prati jednačinu x 2 + y 2 = 1
sin
θ
=
±
1
−
cos
2
θ
and
cos
θ
=
±
1
−
sin
2
θ
.
{\displaystyle \sin \theta =\pm {\sqrt {1-\cos ^{2}\theta }}\quad {\text{and}}\quad \cos \theta =\pm {\sqrt {1-\sin ^{2}\theta }}.\,}
Deljenjem Pitagorinog identiteta sa cos2 θ ili sin2 θ doprinosi stvaranju dva identiteta :
1
+
tan
2
θ
=
sec
2
θ
and
1
+
cot
2
θ
=
csc
2
θ
.
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta \quad {\text{and}}\quad 1+\cot ^{2}\theta =\csc ^{2}\theta .\!}
Korišćenjem ovih identiteta zajedno sa razmernim identitetima, moguće je izraziti bilo koju trigonometrijsku funkciju u izrazima bilo kojih drugih (sve do plus minus znaka) :
Each trigonometric function in terms of the other five.[ 2]
in terms of
sin
θ
{\displaystyle \sin \theta \!}
cos
θ
{\displaystyle \cos \theta \!}
tan
θ
{\displaystyle \tan \theta \!}
csc
θ
{\displaystyle \csc \theta \!}
sec
θ
{\displaystyle \sec \theta \!}
cot
θ
{\displaystyle \cot \theta \!}
sin
θ
=
{\displaystyle \sin \theta =\!}
sin
θ
{\displaystyle \sin \theta \ }
±
1
−
cos
2
θ
{\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}\!}
±
tan
θ
1
+
tan
2
θ
{\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}\!}
1
csc
θ
{\displaystyle {\frac {1}{\csc \theta }}\!}
±
sec
2
θ
−
1
sec
θ
{\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}\!}
±
1
1
+
cot
2
θ
{\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}\!}
cos
θ
=
{\displaystyle \cos \theta =\!}
±
1
−
sin
2
θ
{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}\!}
cos
θ
{\displaystyle \cos \theta \!}
±
1
1
+
tan
2
θ
{\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}\!}
±
csc
2
θ
−
1
csc
θ
{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}\!}
1
sec
θ
{\displaystyle {\frac {1}{\sec \theta }}\!}
±
cot
θ
1
+
cot
2
θ
{\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}\!}
tan
θ
=
{\displaystyle \tan \theta =\!}
±
sin
θ
1
−
sin
2
θ
{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}\!}
±
1
−
cos
2
θ
cos
θ
{\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}\!}
tan
θ
{\displaystyle \tan \theta \!}
±
1
csc
2
θ
−
1
{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}\!}
±
sec
2
θ
−
1
{\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}\!}
1
cot
θ
{\displaystyle {\frac {1}{\cot \theta }}\!}
csc
θ
=
{\displaystyle \csc \theta =\!}
1
sin
θ
{\displaystyle {\frac {1}{\sin \theta }}\!}
±
1
1
−
cos
2
θ
{\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}\!}
±
1
+
tan
2
θ
tan
θ
{\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}\!}
csc
θ
{\displaystyle \csc \theta \!}
±
sec
θ
sec
2
θ
−
1
{\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}\!}
±
1
+
cot
2
θ
{\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}\!}
sec
θ
=
{\displaystyle \sec \theta =\!}
±
1
1
−
sin
2
θ
{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}\!}
1
cos
θ
{\displaystyle {\frac {1}{\cos \theta }}\!}
±
1
+
tan
2
θ
{\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}\!}
±
csc
θ
csc
2
θ
−
1
{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}\!}
sec
θ
{\displaystyle \sec \theta \!}
±
1
+
cot
2
θ
cot
θ
{\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}\!}
cot
θ
=
{\displaystyle \cot \theta =\!}
±
1
−
sin
2
θ
sin
θ
{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}\!}
±
cos
θ
1
−
cos
2
θ
{\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}\!}
1
tan
θ
{\displaystyle {\frac {1}{\tan \theta }}\!}
±
csc
2
θ
−
1
{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}\!}
±
1
sec
2
θ
−
1
{\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}\!}
cot
θ
{\displaystyle \cot \theta \!}
Sve trigonometrijske funkcije ugla θ se mogu konstruisati geometrijski u okviru punog kruga sa centrom u O . Mnogi ovi okviri nisu više u upotrebi. Sinus versus, kosinus versus i eksekant se koristi u navigaciji. Na primer sinus versus formula je korišćena za izračunavanje udaljenosti između dva dela svere. Danas se retko koristi.
Name(s)
Abbreviation(s)
Value[ 3]
versed sine, versine
versin
(
θ
)
{\displaystyle \operatorname {versin} (\theta )}
vers
(
θ
)
{\displaystyle \operatorname {vers} (\theta )}
ver
(
θ
)
{\displaystyle \operatorname {ver} (\theta )}
1
−
cos
(
θ
)
{\displaystyle 1-\cos(\theta )}
versed cosine, vercosine
vercosin
(
θ
)
{\displaystyle \operatorname {vercosin} (\theta )}
1
+
cos
(
θ
)
{\displaystyle 1+\cos(\theta )}
coversed sine, coversine
coversin
(
θ
)
{\displaystyle \operatorname {coversin} (\theta )}
cvs
(
θ
)
{\displaystyle \operatorname {cvs} (\theta )}
1
−
sin
(
θ
)
{\displaystyle 1-\sin(\theta )}
coversed cosine, covercosine
covercosin
(
θ
)
{\displaystyle \operatorname {covercosin} (\theta )}
1
+
sin
(
θ
)
{\displaystyle 1+\sin(\theta )}
half versed sine, haversine
haversin
(
θ
)
{\displaystyle \operatorname {haversin} (\theta )}
1
−
cos
(
θ
)
2
{\displaystyle {\frac {1-\cos(\theta )}{2}}}
half versed cosine, havercosine
havercosin
(
θ
)
{\displaystyle \operatorname {havercosin} (\theta )}
1
+
cos
(
θ
)
2
{\displaystyle {\frac {1+\cos(\theta )}{2}}}
half coversed sine, hacoversine
hacoversin
(
θ
)
{\displaystyle \operatorname {hacoversin} (\theta )}
1
−
sin
(
θ
)
2
{\displaystyle {\frac {1-\sin(\theta )}{2}}}
half coversed cosine, hacovercosine
hacovercosin
(
θ
)
{\displaystyle \operatorname {hacovercosin} (\theta )}
1
+
sin
(
θ
)
2
{\displaystyle {\frac {1+\sin(\theta )}{2}}}
exterior secant, exsecant
exsec
(
θ
)
{\displaystyle \operatorname {exsec} (\theta )}
sec
(
θ
)
−
1
{\displaystyle \sec(\theta )-1}
exterior cosecant, excosecant
excsc
(
θ
)
{\displaystyle \operatorname {excsc} (\theta )}
csc
(
θ
)
−
1
{\displaystyle \csc(\theta )-1}
chord
crd
(
θ
)
{\displaystyle \operatorname {crd} (\theta )}
2
sin
θ
2
{\displaystyle 2\sin {\frac {\theta }{2}}}
Ispitivanjem punog kruga, prateća svojstva trigonometrijskih funkcija mogu biti utvrđeni.
Kada su trigonometrijske funkcije reflektovane na određen ugao, rezultat je često jedna od trigonometrijskih funkcija. To nas vodi do sledećih identiteta :
Reflected in
θ
=
0
{\displaystyle \theta =0}
[ 4]
Reflected in
θ
=
π
/
2
{\displaystyle \theta =\pi /2}
[ 5]
Reflected in
θ
=
π
{\displaystyle \theta =\pi }
sin
(
−
θ
)
=
−
sin
θ
cos
(
−
θ
)
=
+
cos
θ
tan
(
−
θ
)
=
−
tan
θ
csc
(
−
θ
)
=
−
csc
θ
sec
(
−
θ
)
=
+
sec
θ
cot
(
−
θ
)
=
−
cot
θ
{\displaystyle {\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=+\cos \theta \\\tan(-\theta )&=-\tan \theta \\\csc(-\theta )&=-\csc \theta \\\sec(-\theta )&=+\sec \theta \\\cot(-\theta )&=-\cot \theta \\\end{aligned}}}
sin
(
π
2
−
θ
)
=
+
cos
θ
cos
(
π
2
−
θ
)
=
+
sin
θ
tan
(
π
2
−
θ
)
=
+
cot
θ
csc
(
π
2
−
θ
)
=
+
sec
θ
sec
(
π
2
−
θ
)
=
+
csc
θ
cot
(
π
2
−
θ
)
=
+
tan
θ
{\displaystyle {\begin{aligned}\sin({\tfrac {\pi }{2}}-\theta )&=+\cos \theta \\\cos({\tfrac {\pi }{2}}-\theta )&=+\sin \theta \\\tan({\tfrac {\pi }{2}}-\theta )&=+\cot \theta \\\csc({\tfrac {\pi }{2}}-\theta )&=+\sec \theta \\\sec({\tfrac {\pi }{2}}-\theta )&=+\csc \theta \\\cot({\tfrac {\pi }{2}}-\theta )&=+\tan \theta \\\end{aligned}}}
sin
(
π
−
θ
)
=
+
sin
θ
cos
(
π
−
θ
)
=
−
cos
θ
tan
(
π
−
θ
)
=
−
tan
θ
csc
(
π
−
θ
)
=
+
csc
θ
sec
(
π
−
θ
)
=
−
sec
θ
cot
(
π
−
θ
)
=
−
cot
θ
{\displaystyle {\begin{aligned}\sin(\pi -\theta )&=+\sin \theta \\\cos(\pi -\theta )&=-\cos \theta \\\tan(\pi -\theta )&=-\tan \theta \\\csc(\pi -\theta )&=+\csc \theta \\\sec(\pi -\theta )&=-\sec \theta \\\cot(\pi -\theta )&=-\cot \theta \\\end{aligned}}}
Pod smenom funkcije kruga nekim određenim uglom često je moguće uočiti različite trigonometrijske funkcije koje pokazuju te rezultate u jednostavnijem obliku. Neki primeri ovoga su prikazani smenom funkcija kruga sa π/2, π i 2π radijana. Zbog stila funkcija je π ili 2π, ima slučajeva kada je nova funkcija u potpunosti ista kao stara bez smene.
Shift by π/2
Shift by π [ 6]
Shift by 2π [ 7]
sin
(
θ
+
π
2
)
=
+
cos
θ
cos
(
θ
+
π
2
)
=
−
sin
θ
tan
(
θ
+
π
2
)
=
−
cot
θ
csc
(
θ
+
π
2
)
=
+
sec
θ
sec
(
θ
+
π
2
)
=
−
csc
θ
cot
(
θ
+
π
2
)
=
−
tan
θ
{\displaystyle {\begin{aligned}\sin(\theta +{\tfrac {\pi }{2}})&=+\cos \theta \\\cos(\theta +{\tfrac {\pi }{2}})&=-\sin \theta \\\tan(\theta +{\tfrac {\pi }{2}})&=-\cot \theta \\\csc(\theta +{\tfrac {\pi }{2}})&=+\sec \theta \\\sec(\theta +{\tfrac {\pi }{2}})&=-\csc \theta \\\cot(\theta +{\tfrac {\pi }{2}})&=-\tan \theta \end{aligned}}}
sin
(
θ
+
π
)
=
−
sin
θ
cos
(
θ
+
π
)
=
−
cos
θ
tan
(
θ
+
π
)
=
+
tan
θ
csc
(
θ
+
π
)
=
−
csc
θ
sec
(
θ
+
π
)
=
−
sec
θ
cot
(
θ
+
π
)
=
+
cot
θ
{\displaystyle {\begin{aligned}\sin(\theta +\pi )&=-\sin \theta \\\cos(\theta +\pi )&=-\cos \theta \\\tan(\theta +\pi )&=+\tan \theta \\\csc(\theta +\pi )&=-\csc \theta \\\sec(\theta +\pi )&=-\sec \theta \\\cot(\theta +\pi )&=+\cot \theta \\\end{aligned}}}
sin
(
θ
+
2
π
)
=
+
sin
θ
cos
(
θ
+
2
π
)
=
+
cos
θ
tan
(
θ
+
2
π
)
=
+
tan
θ
csc
(
θ
+
2
π
)
=
+
csc
θ
sec
(
θ
+
2
π
)
=
+
sec
θ
cot
(
θ
+
2
π
)
=
+
cot
θ
{\displaystyle {\begin{aligned}\sin(\theta +2\pi )&=+\sin \theta \\\cos(\theta +2\pi )&=+\cos \theta \\\tan(\theta +2\pi )&=+\tan \theta \\\csc(\theta +2\pi )&=+\csc \theta \\\sec(\theta +2\pi )&=+\sec \theta \\\cot(\theta +2\pi )&=+\cot \theta \end{aligned}}}
Ilustracija adicione formule za sinus i kosinus. Ilustracija adicione formule za tangens. Poznate su kao adicione i oduzimajuće teoreme ili formule.
One potiču iz desetog veka i utvrdio ih je persijski matematicar Abū al-Wafā' Būzjānī .
Jedan metod dokazivanja ovih identiteta se poklapa sa Eulerovom formulom.
Za diagram adicije ugla za sinus i kosinus, tamna linija sa 1 svoje dužine je dužine jedan. Hipotenuza desnog ugla trougla sa uglom β sa kojim daje sinus β i kosinus β. Kosinus β linija je hipotenuza desnog ugla trougla sa uglom α tako da ima sa strane sinus α i kosinus α i oboje pomoženo sa kosinus β. Ovo je isto za sinus β liniju.
Uopsteno dijagram može biti korišćen da pokaže sinus i kosinus zbira identiteta
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
{\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }
cos
(
α
+
β
)
=
cos
α
cos
β
−
sin
α
sin
β
{\displaystyle \cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta }
Jer suprotne strane pravougaonika su jednake.
Sine
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 \!}
[ 8] [ 9]
Cosine
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 \,}
[ 9] [ 10]
Tangent
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 }}}
[ 9] [ 11]
Arcsine
arcsin
α
±
arcsin
β
=
arcsin
(
α
1
−
β
2
±
β
1
−
α
2
)
{\displaystyle \arcsin \alpha \pm \arcsin \beta =\arcsin \left(\alpha {\sqrt {1-\beta ^{2}}}\pm \beta {\sqrt {1-\alpha ^{2}}}\right)}
[ 12]
Arccosine
arccos
α
±
arccos
β
=
arccos
(
α
β
∓
(
1
−
α
2
)
(
1
−
β
2
)
)
{\displaystyle \arccos \alpha \pm \arccos \beta =\arccos \left(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}}\right)}
[ 13]
Arctangent
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)}
[ 14]
Zbir i razlika formula sinusa i kosinusa može biti napisana u matriks formi kao :
(
cos
α
−
sin
α
sin
α
cos
α
)
(
cos
β
−
sin
β
sin
β
cos
β
)
=
(
cos
α
cos
β
−
sin
α
sin
β
−
cos
α
sin
β
−
sin
α
cos
β
sin
α
cos
β
+
cos
α
sin
β
−
sin
α
sin
β
+
cos
α
cos
β
)
=
(
cos
(
α
+
β
)
−
sin
(
α
+
β
)
sin
(
α
+
β
)
cos
(
α
+
β
)
)
.
{\displaystyle {\begin{aligned}&{}\quad \left({\begin{array}{rr}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{rr}\cos \beta &-\sin \beta \\\sin \beta &\cos \beta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos \alpha \cos \beta -\sin \alpha \sin \beta &-\cos \alpha \sin \beta -\sin \alpha \cos \beta \\\sin \alpha \cos \beta +\cos \alpha \sin \beta &-\sin \alpha \sin \beta +\cos \alpha \cos \beta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos(\alpha +\beta )&-\sin(\alpha +\beta )\\\sin(\alpha +\beta )&\cos(\alpha +\beta )\end{array}}\right).\end{aligned}}}
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)}
U ova dva identiteta asimetričnost se pojavljuje ali nije viđena u slučaju zbira konačnosti mnogih uslova : U svakom produktu, ima nekoliko konačnih sinus faktora i dvosmislenosti mnogih kosinus faktora.
Samo ako beskonacnost mnogih ovih uslova θi
ako e k k = 0, 1, 2, 3, ...) bude k ti stepen elementarnog simetričnog polinoma u varijablama
x
i
=
tan
θ
i
{\displaystyle x_{i}=\tan \theta _{i}\,}
for i = 0, 1, 2, 3, ..., i.e.,
e
0
=
1
e
1
=
∑
i
x
i
=
∑
i
tan
θ
i
e
2
=
∑
i
<
j
x
i
x
j
=
∑
i
<
j
tan
θ
i
tan
θ
j
e
3
=
∑
i
<
j
<
k
x
i
x
j
x
k
=
∑
i
<
j
<
k
tan
θ
i
tan
θ
j
tan
θ
k
⋮
⋮
{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&{}\ \ \vdots &&{}\ \ \vdots \end{aligned}}}
Onda
tan
(
∑
i
θ
i
)
=
e
1
−
e
3
+
e
5
−
⋯
e
0
−
e
2
+
e
4
−
⋯
.
{\displaystyle \tan \left(\sum _{i}\theta _{i}\right)={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots }}.\!}
Broj uslova sa desne strane zavisi od broja uslova sa leve.
Na primer:
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}}},\\[8pt]\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})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\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.
sec
(
∑
i
θ
i
)
=
∏
i
sec
θ
i
e
0
−
e
2
+
e
4
−
⋯
csc
(
∑
i
θ
i
)
=
∏
i
sec
θ
i
e
1
−
e
3
+
e
5
−
⋯
{\displaystyle {\begin{aligned}\sec \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]\csc \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}
gde e k k ti stepen elementarnog simetričnog polinoma u n varijabli x i θ i i = 1, ..., n , i broj uslova u imeniocu i broj faktora u rezultatu u broiocu zavisi od broja uslova u zbiru sa leve strane. Slučaj konačnosti mnogih uslova može biti proverena matematičkom indukcijom. Konvergencija serija u imeniocu može biti pokazana pisanjem sekans identiteta u formi
e
0
−
e
2
+
e
4
−
⋯
=
∏
i
sec
θ
i
sec
(
∑
i
θ
i
)
{\displaystyle e_{0}-e_{2}+e_{4}-\cdots ={\frac {\prod _{i}\sec \theta _{i}}{\sec \left(\sum _{i}\theta _{i}\right)}}}
i onda posmatra da je leva strana konvergentna ukoliko je desna strana konvergentna, i slična kosenkans identitetu.
Na primer,
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}}}
Function
Inverse function[ 15]
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
−
i
1
−
x
2
)
{\displaystyle \arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\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\,}
^ Schaumberger, N. "A Classroom Theorem on Trigonometric Irrationalities." Two-Year College Math. J. 5, 73-76, 1974. also see Weisstein, Eric W. "Niven's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NivensTheorem.html
^ Abramowitz & Stegun. pp. 73, 4.3.45
^ Abramowitz & Stegun. pp. 78, 4.3.147
^ Abramowitz & Stegun. pp. 72, 4.3.13–15
^ „The Elementary Identities[[Kategorija:Botovski naslovi]]” . Arhivirano iz originala 30. 07. 2017. g. Pristupljeno 02. 04. 2014 . ^ Abramowitz & Stegun. pp. 72, 4.3.9
^ Abramowitz & Stegun. pp. 72, 4.3.7–8
^ Abramowitz & Stegun. pp. 72, 4.3.16
^ a b v Weisstein, Eric W. „Trigonometric Addition Formulas” . MathWorld ^ Abramowitz & Stegun. pp. 72, 4.3.17
^ Abramowitz & Stegun. pp. 72, 4.3.18
^ Abramowitz & Stegun. pp. 80, 4.4.42
^ Abramowitz & Stegun. pp. 80, 4.4.43
^ Abramowitz & Stegun. pp. 80, 4.4.36
^ Abramowitz & Stegun. pp. 80, 4.4.26–31
![{\displaystyle {\begin{aligned}&{}\quad \left({\begin{array}{rr}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{rr}\cos \beta &-\sin \beta \\\sin \beta &\cos \beta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos \alpha \cos \beta -\sin \alpha \sin \beta &-\cos \alpha \sin \beta -\sin \alpha \cos \beta \\\sin \alpha \cos \beta +\cos \alpha \sin \beta &-\sin \alpha \sin \beta +\cos \alpha \cos \beta \end{array}}\right)\\[12pt]&=\left({\begin{array}{rr}\cos(\alpha +\beta )&-\sin(\alpha +\beta )\\\sin(\alpha +\beta )&\cos(\alpha +\beta )\end{array}}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbbc83690fbaa9724724144f29919d3ad4f9d31c)
![{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&{}\ \ \vdots &&{}\ \ \vdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe5c7c43e0176206f134067e005328cb0a31cce)
![{\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}}},\\[8pt]\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})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9196b8ab5e0aef6cf7d7fec7d7546f2a6e7d2f)
![{\displaystyle {\begin{aligned}\sec \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]\csc \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d12744003bcf5becf301096c3d12ca1d1e1055)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e082e69aab7a9c782f590f3a93262ba3101899b7)
