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 = 2π {\displaystyle \pi } 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
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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
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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
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Ostale funkcije korištene u prošlosti
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Simetrija, pomak i periodičnost
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Zbroj i razlika kutova
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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
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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
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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
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Neka je e k {\displaystyle e_{k}\,} 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
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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 − 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
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Tn je n -ti Čebiševljev polinom
cos n θ = T n ( cos θ ) {\displaystyle \cos n\theta =T_{n}(\cos \theta )\,} 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}\,} [14]
Trigonomterijske jednakosti dvostrukih, trostrukih i polovičnih kutova
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Sinus, kosinus i tangens višestrukih kutova
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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
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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
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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
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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
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Formule pretvorbi umnoška u zbroj i zbroja u umnožak
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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
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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
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Charles Hermite je pokazao da vrijedi određena jednakost[20] 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
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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
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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]
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
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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
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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
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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 }.\,} Jednakosti inverznih trigonometrijskih funkcija
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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
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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
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Povezanost s beskonačnim produktima
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Jednakosti bez varijabli
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Infinitezimalni račun
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Derivacije
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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
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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
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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 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
↑ 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
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Podrobniji članak o temi: Kut
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/852d1e76b4190362dde8aefd41ddca69347d1961)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2c5db750b9b1fba93aa9fefa0a792dd377829e)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c82ca2d8da55d938d78aad99295a11f79665fde)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1feb844ca95b6583a9ded365961e15f0a0b41b8d)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3414d47dcb384f6ef6779a03fc3f6bf78c2277)
![{\displaystyle \sin[\arccos(x)]={\sqrt {1-x^{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89020e4173e7c25657019ade3629e832bb572ff7)
![{\displaystyle \operatorname {tg} [\arcsin(x)]={\frac {x}{\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/987e9d7cbe18dedec623c950d60cae9ae96b31e1)
![{\displaystyle \sin[\operatorname {arctg} (x)]={\frac {x}{\sqrt {1+x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccb88e6d593826b440868aad7289e3909b16b15)
![{\displaystyle \operatorname {tg} [\arccos(x)]={\frac {\sqrt {1-x^{2}}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d703d7e114ec7e5bbec1392c7af66fc32cf825ae)
![{\displaystyle \cos[\operatorname {arctg} (x)]={\frac {1}{\sqrt {1+x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69d9c32c169fdc562ee70c186a7db504aa68d52a)
![{\displaystyle \operatorname {ctg} [\arcsin(x)]={\frac {\sqrt {1-x^{2}}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfe4684b2e7daa5972ddc23820924b720fccc9a)
![{\displaystyle \cos[\arcsin(x)]={\sqrt {1-x^{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6123652c97881e3910234616ea5ee831deae4a73)
![{\displaystyle \operatorname {ctg} [\arccos(x)]={\frac {x}{\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0f033513b2f0fd1adb738b46225e3070ce41e8)
![{\displaystyle |\sin x|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\operatorname {tg} \left(2^{n}x\right)\right|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/897fe4518be9ba1bb0234ee4cf9f91ea14bbb1ed)
![{\displaystyle {\begin{aligned}&\cos \left({\frac {2\pi }{21}}\right)+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)\\[10pt]&{}\qquad {}+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15e07fe52f1162c9c933e28a6bec4fb9f59528e8)
