Tożsamości trygonometryczne – podstawowe zależności pomiędzy funkcjami trygonometrycznymi.
Tożsamości pitagorejskie
Osobny artykuł: jedynka trygonometryczna.
Wzór
![{\displaystyle \sin ^{2}x+\cos ^{2}x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1938e6828e597c076248d3ec430e0a7e5f98c8)
jest prawdziwy dla dowolnej liczby rzeczywistej (a nawet zespolonej, przy przyjęciu ogólniejszych definicji). Tożsamość ta uznawana jest za podstawową tożsamość trygonometryczną. Zwana często jedynką trygonometryczną bądź trygonometrycznym twierdzeniem Pitagorasa.
Istnieją również dwie inne wariacje tego wzoru:
![{\displaystyle \sec ^{2}x-\operatorname {tg} ^{2}\ x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3de7f4f768a1ca30503b3c3a5f0fafe9b70e56cf)
![{\displaystyle \operatorname {cosec} ^{2}x-\operatorname {ctg} ^{2}\ x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4a197e86ca9987a3ee6a3c7f3f14cd5a174178)
Okresowość funkcji
Funkcje trygonometryczne są okresowe
![{\displaystyle {\begin{array}{l}\sin x=\sin(x+2k\pi )&\operatorname {tg} x=\operatorname {tg} (x+k\pi )\\\cos x=\cos(x+2k\pi )&\operatorname {ctg} x=\operatorname {ctg} (x+k\pi )\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85fb91c907a867bc256a4076830292f44e21a9dc)
Definicje tangensa i cotangensa
![{\displaystyle \operatorname {tg} x={\frac {\sin x}{\cos x}},\quad {\text{dla }}x\neq {\frac {\pi }{2}}+k\pi ,\quad {\text{gdzie k}}\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca81062853c82df0c0691ff1cdd68c27f44c7321)
![{\displaystyle \operatorname {ctg} x={\frac {\cos x}{\sin x}},\quad {\text{dla }}x\neq k\pi ,\quad {\text{gdzie k}}\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ea604499b970ff5c1fc29f9c4cb5ac2e896ad6)
![{\displaystyle \lim _{x\to x_{0}^{\pm }}~{\operatorname {ctg} x}=\lim _{x\to x_{0}^{\pm }}~{\frac {1}{\operatorname {tg} x}},\quad {\text{dla }}x_{0}\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ffc6dd79960e63e3d422c99874ed996354ef95)
Przedstawienia przy pomocy funkcji cosinus
![{\displaystyle |\sin x|={\sqrt {1-\cos ^{2}x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a47b5ff227c046ddaa56d6a0853daec7027b4413)
![{\displaystyle |\operatorname {tg} x|={\frac {|\sin x|}{|\cos x|}}={\frac {\sqrt {1-\cos ^{2}x}}{|\cos x|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee92d3a42ceeacf4f0f3d1efb5612ed638a4cb23)
![{\displaystyle |\operatorname {ctg} x|={\frac {|\cos x|}{|\sin x|}}={\frac {|\cos x|}{\sqrt {1-\cos ^{2}x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8379eb0ff0f4aea1aa3d6bd9b408754be658bc26)
Przedstawienia przy pomocy funkcji sinus
![{\displaystyle |\cos x|={\sqrt {1-\sin ^{2}x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47cbce1242fe88da82ee8101f1ffce082e5aba6e)
![{\displaystyle |\operatorname {tg} x|={\frac {|\sin x|}{|\cos x|}}={\frac {|\sin x|}{\sqrt {1-\sin ^{2}x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47cd18fb03d51c7555ecfb161faa778b456f6aae)
![{\displaystyle |\operatorname {ctg} x|={\frac {|\cos x|}{|\sin x|}}={\frac {\sqrt {1-\sin ^{2}x}}{|\sin x|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415b76676c8411832b56fe86fcc32d89b5ad88c0)
Parzystość i nieparzystość funkcji trygonometrycznych
![{\displaystyle {\begin{array}{l}\sin(-x)=-\sin x&\operatorname {tg} (-x)=-\operatorname {tg} x\\\cos(-x)=\cos x&\operatorname {ctg} (-x)=-\operatorname {ctg} x\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43c34052d2e336bc3691d527c1374cdee6716e5d)
Zależności pomiędzy funkcjami a kofunkcjami
Równości
![{\displaystyle {\begin{aligned}&\sin x=\cos \left({\frac {\pi }{2}}-x\right)&&\cos x=\sin \left({\frac {\pi }{2}}-x\right)\\[.2em]&\operatorname {tg} x=\operatorname {ctg} \left({\frac {\pi }{2}}-x\right)&&\operatorname {ctg} x=\operatorname {tg} \left({\frac {\pi }{2}}-x\right)\\[.2em]&\sec x=\operatorname {cosec} \left({\frac {\pi }{2}}-x\right)&&\operatorname {cosec} x=\sec \left({\frac {\pi }{2}}-x\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd95bdb88e651940e908933f83dcad199334479)
nazywa się związkami pomiędzy funkcjami a ich kofunkcjami. Kofunkcją sinusa jest cosinus, cosinusa sinus, tangensa cotangens itd.
Odwrotności
Funkcje trygonometryczne można układać w pary według kofunkcji lub według odwrotności. Odwrotnością sinusa jest cosecans, cosinusa secans, tangensa cotangens (i oczywiście na odwrót):
![{\displaystyle {\begin{aligned}&\sin x={\frac {1}{\operatorname {cosec} x}}&&\operatorname {cosec} x={\frac {1}{\sin x}}\\[.5em]&\cos x={\frac {1}{\sec x}}&&\sec x={\frac {1}{\cos x}}\\[.5em]&\operatorname {tg} x={\frac {1}{\operatorname {ctg} x}}&&\operatorname {ctg} x={\frac {1}{\operatorname {tg} x}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c997a622134bd906219470948bb3121aa2633ce5)
Funkcje sumy i różnicy kątów
![{\displaystyle \sin(x\pm y)=\sin x\cdot \cos y\pm \cos x\cdot \sin y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6586aa07fe77e9ccd8617dbd587397a0ba025fe2)
![{\displaystyle \cos(x\pm y)=\cos x\cos y\mp \sin x\sin y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde27654b87805940cfef57359d550d843c7abba)
![{\displaystyle \operatorname {tg} (x\pm y)={\frac {\operatorname {tg} x\pm \operatorname {tg} y}{1\mp \operatorname {tg} x\operatorname {tg} y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55f7105d653a31c2a1fe30a0b244e47032160937)
![{\displaystyle \operatorname {ctg} (x\pm y)={\frac {\operatorname {ctg} x\cdot \operatorname {ctg} y\mp 1}{\operatorname {ctg} y\pm \operatorname {ctg} x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77f7d1a22a8c72830ab0b1216ea20804ec2da5ce)
Funkcje wielokrotności kątów
Wzory na dwukrotność kąta otrzymuje się przez podstawienie
we wzorach na funkcje sumy kątów.
![{\displaystyle \sin 2x=2\sin x\cos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2433b44a620c864ff755594ae13751ece26ea9)
![{\displaystyle \cos 2x=\cos ^{2}x-\sin ^{2}x=1-2\sin ^{2}x=2\cos ^{2}x-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4411eb657d837435eb68f9e5a8d1401b5104a859)
![{\displaystyle \operatorname {tg} 2x={\frac {2\operatorname {tg} x}{1-\operatorname {tg} ^{2}\ x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc6b2dd53995ead57e54d75a2ff8900fb7ccf65)
![{\displaystyle \operatorname {ctg} 2x={\frac {\operatorname {ctg} x-\operatorname {tg} x}{2}}={\frac {\operatorname {ctg} ^{2}\ x-1}{2\operatorname {ctg} x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9b445c5d04582db26c8319d321e2793ef04f77)
![{\displaystyle \sin 3x=3\sin x-4\sin ^{3}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/617e32b3f98d83f5baaaacbf2ea995a02b98240e)
![{\displaystyle \cos 3x=4\cos ^{3}x-3\cos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7326ed2f260bf253e2e478c3b8f10d2b4b7a7ba8)
![{\displaystyle \operatorname {tg} 3x={\frac {3\operatorname {tg} x-\operatorname {tg} ^{3}\ x}{1-3\operatorname {tg} ^{2}\ x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5829d3a6f05be327a7762007981926eafe28407)
![{\displaystyle \operatorname {ctg} 3x={\frac {\operatorname {ctg} ^{3}\ x-3\operatorname {ctg} x}{3\operatorname {ctg} ^{2}\ x-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc0a8c0414633afc6fb484655e0e6c3f4fb2c0b)
![{\displaystyle \sin 4x=8\cos ^{3}x\sin x-4\cos x\sin x=4\cos ^{3}x\sin x-4\cos x\sin ^{3}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d5f7d51397ff8427cd652473cb28dbd893f981)
![{\displaystyle \cos 4x=8\cos ^{4}x-8\cos ^{2}x+1=8\sin ^{4}x-8\sin ^{2}x+1=\cos ^{4}x-6\cos ^{2}x\sin ^{2}x+\sin ^{4}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f979481c89a3665c6017a50fc97a166e5fbb45d4)
![{\displaystyle \operatorname {tg} 4x={\frac {4\operatorname {tg} x-4\operatorname {tg} ^{3}\ x}{1-6\operatorname {tg} ^{2}\ x+\operatorname {tg} ^{4}\ x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a0b34d050a2fbaddb710a7a2751ddeb78f2527)
![{\displaystyle \operatorname {ctg} 4x={\frac {\operatorname {ctg} ^{4}\ x-6\operatorname {ctg} ^{2}\ x+1}{4\operatorname {ctg} ^{3}\ x-4\operatorname {ctg} x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3eb92a57e0ce574ab36c0f1cf67593a2668b05)
Ogólnie:
![{\displaystyle {\begin{aligned}\sin nx&=\sum _{i=0}^{\infty }(-1)^{i}\cdot {n \choose 2i+1}\cos ^{n-2i-1}x\sin ^{2i+1}x\\[2pt]&=\sum _{i=0}^{\lfloor {\frac {n}{2}}\rfloor }(-1)^{i}\cdot {n \choose 2i+1}\cos ^{n-2i-1}x\sin ^{2i+1}x\\&=n\cos ^{n-1}x\sin x-{n \choose 3}\cos ^{n-3}x\sin ^{3}x+{n \choose 5}\cos ^{n-5}x\sin ^{5}x-{n \choose 7}\cos ^{n-7}x\sin ^{7}x+\dots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/714e1218d101c47c2664961856dd7d3326ba7738)
![{\displaystyle {\begin{aligned}\cos nx&=\sum _{i=0}^{\infty }(-1)^{i}\cdot {n \choose 2i}\cos ^{n-2i}x\sin ^{2i}x\\[2pt]&=\sum _{i=0}^{\lfloor {\frac {n}{2}}\rfloor }(-1)^{i}\cdot {n \choose 2i}\cos ^{n-2i}x\sin ^{2i}x\\&=\cos ^{n}x-{n \choose 2}\cos ^{n-2}x\sin ^{2}x+{n \choose 4}\cos ^{n-4}x\sin ^{4}x-{n \choose 6}\cos ^{n-6}x\sin ^{6}x+\dots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb40d217ed38a880497203d0468cb9a544202f5)
![{\displaystyle \operatorname {tg} nx=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{n \choose 2i+1}\operatorname {tg} ^{2i+1}x\cdot (-1)^{i}\cdot \left(\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{n \choose 2i}\operatorname {tg} ^{2i}x\cdot (-1)^{i}\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7482749c9089beffdd47a60d3404c231c8f4515b)
Funkcje kąta połówkowego
![{\displaystyle \left|\sin {\frac {1}{2}}x\right|={\sqrt {\frac {1-\cos x}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc79f93f23bbf8bacd7c46a3b0c132e87be0064)
![{\displaystyle \left|\cos {\frac {1}{2}}x\right|={\sqrt {\frac {1+\cos x}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e55fc670d328f8a09163f59cd464e56bd4624165)
![{\displaystyle \left|\operatorname {tg} {\frac {1}{2}}x\right|={\sqrt {\frac {1-\cos x}{1+\cos x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c13e06528cc28bc67befa44f7b0d9ab390adeb)
![{\displaystyle \operatorname {tg} {\frac {1}{2}}x={\frac {1-\cos x}{\sin x}}={\frac {\sin x}{1+\cos x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dcd61accf32c34f2018c4911557160541132cdd)
![{\displaystyle \left|\operatorname {ctg} {\frac {1}{2}}x\right|={\sqrt {\frac {1+\cos x}{1-\cos x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/963c2668f997b765ea8c7b2d6c41c5d66b12bbfd)
![{\displaystyle \operatorname {ctg} {\frac {1}{2}}x={\frac {1+\cos x}{\sin x}}={\frac {\sin x}{1-\cos x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3187f6dd75b5375893fa0a6935528deecc879c)
Suma i różnica funkcji
![{\displaystyle \sin x\pm \sin y=2\sin {\frac {x\pm y}{2}}\cdot \cos {\frac {x\mp y}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/782c9d591124c7f137deb6a4fb42ec71c7ac81f3)
![{\displaystyle \cos x+\cos y=2\cos {\frac {x+y}{2}}\cdot \cos {\frac {x-y}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0b11032bca6ee015ff054b6c7a54616ee9cd78)
![{\displaystyle \cos x-\cos y=-2\sin {\frac {x+y}{2}}\cdot \sin {\frac {x-y}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/632b87b0ed727edd2cedc1dde7b4236c044239c2)
![{\displaystyle \operatorname {tg} x\pm \operatorname {tg} y={\frac {\sin(x\pm y)}{\cos x\cos y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c76a8dcc58bce082a64473c101f93c05508a1a85)
![{\displaystyle \operatorname {tg} x+\operatorname {ctg} y={\frac {\cos(x-y)}{\cos x\sin y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7861843a4069882b4963b737f1a1477a54d73f)
![{\displaystyle \operatorname {ctg} x\pm \operatorname {ctg} y={\frac {\sin(y\pm x)}{\sin x\sin y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd788ddae610475c7d18036922a7af991d76748)
![{\displaystyle \operatorname {ctg} x-\operatorname {tg} y={\frac {\cos(x+y)}{\sin x\cos y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77aa105493b988fc8431f243e38c284b5ca65f91)
![{\displaystyle 1-\cos x=2\sin ^{2}{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa15d050574af12a0f49af27b12a543e40fbe6f6)
![{\displaystyle 1+\cos x=2\cos ^{2}{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b84d8a2a7498fa8d2048f5d6aa78de729c12e7)
![{\displaystyle 1-\sin x=2\sin ^{2}\left({\frac {1}{4}}\pi -{\frac {1}{2}}x\right)=2\cos ^{2}\left({\frac {1}{4}}\pi +{\frac {1}{2}}x\right)=\left(\sin {\frac {x}{2}}-\cos {\frac {x}{2}}\right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e5e7e82c88f735defe3503078bccd06a81e457)
![{\displaystyle 1+\sin x=2\cos ^{2}\left({\frac {1}{4}}\pi -{\frac {1}{2}}x\right)=2\sin ^{2}\left({\frac {1}{4}}\pi +{\frac {1}{2}}x\right)=\left(\sin {\frac {x}{2}}+\cos {\frac {x}{2}}\right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c36fc3752ddd661f34c56db5ff78fb095c2fa44)
Iloczyn w postaci sumy
![{\displaystyle \cos x\cdot \cos y={\frac {\cos(x-y)+\cos(x+y)}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77bbed94232049dcf3316818fdc7c73bc42dff56)
![{\displaystyle \sin x\cdot \sin y={\frac {\cos(x-y)-\cos(x+y)}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75738aba79de144a2294babc77ef155ce99decf4)
![{\displaystyle \sin x\cdot \cos y={\frac {\sin(x-y)+\sin(x+y)}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/531ffe97790625282f9cfc351f8fb2f20fe3c65c)
![{\displaystyle \sin x\cdot \sin y\cdot \sin z={\frac {\sin(x+y-z)+\sin(y+z-x)+\sin(z+x-y)-\sin(x+y+z)}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6d2a37be87b2c74cf17bb2666e6ba0f33bf2c1)
![{\displaystyle \sin x\cdot \sin y\cdot \cos z={\frac {-\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)-\cos(x+y+z)}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cab0e009757ae13d4501fcdcab69b7b97cbce6a)
![{\displaystyle \sin x\cdot \cos y\cdot \cos z={\frac {\sin(x+y-z)-\sin(y+z-x)+\sin(z+x-y)+\sin(x+y+z)}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb6278b773080e282c0144ebd4ebe3ea2819717)
![{\displaystyle \cos x\cdot \cos y\cdot \cos z={\frac {\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)+\cos(x+y+z)}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54a0e1cf728937b234f58a2c4d7bbe825fd02ab)
Potęgi w postaci sumy
![{\displaystyle \sin ^{2}x={\frac {1-\cos 2x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3436f66943e84b9f8a51509acba59992807951)
![{\displaystyle \cos ^{2}x={\frac {1+\cos 2x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b020bd1a303761fd09c54958093adf61be8768e)
![{\displaystyle \sin ^{2}x\cos ^{2}x={\frac {1-\cos 4x}{8}}={\frac {\sin ^{2}2x}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c72e69329f365527db2074132b13a71622e1a4c3)
![{\displaystyle \sin ^{3}x={\frac {3\sin x-\sin 3x}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ec7f7378a2ef47e3783bca8a3bc7b818418bbc)
![{\displaystyle \cos ^{3}x={\frac {3\cos x+\cos 3x}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908aee93a18f265ba5992c718a2606780db29fdf)
![{\displaystyle \sin ^{4}x={\frac {\cos 4x-4\cos 2x+3}{8}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dab101a26610b15e8d3f29e5ae417506b2c49888)
![{\displaystyle \cos ^{4}x={\frac {\cos 4x+4\cos 2x+3}{8}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/739bafc8ec3673747911f8f0245a32bcecb79cbf)
![{\displaystyle \sin ^{2}x-\sin ^{2}y=\sin(x+y)\cdot \sin(x-y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ab6203763d8c9175af1c6c98afe04dadc52ba5)
Funkcje trygonometryczne wyrażone przy pomocy tangensa połowy kąta
![{\displaystyle \sin x={\frac {2\operatorname {tg} {\frac {x}{2}}}{1+\operatorname {tg} ^{2}{\frac {x}{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2051c6c6fe02cc26528f602d09fb98209934f321)
![{\displaystyle \cos x={\frac {1-\operatorname {tg} ^{2}{\frac {x}{2}}}{1+\operatorname {tg} ^{2}{\frac {x}{2}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de4a2655c92f3bafb9215f9f1bf16e7c5868b13f)
![{\displaystyle \operatorname {tg} x={\frac {2\operatorname {tg} {\frac {x}{2}}}{1-\operatorname {tg} ^{2}{\frac {x}{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2835531c4d9668e570c6549464d60bff531535de)
Powyższe tożsamości znalazły zastosowanie w tzw. podstawieniu uniwersalnym, stosowanym przy obliczaniu całek typu
gdzie
jest funkcją wymierną zmiennych
Stosuje się podstawienie:
![{\displaystyle \operatorname {tg} {\frac {x}{2}}=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7aed887d093b933c2ea493fac4c83b3fcabaaee)
![{\displaystyle x=2\operatorname {arctg} \;t+2k\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc37f77cd277afacabf8c57072db2d3ffc799036)
![{\displaystyle dx={\frac {2}{1+t^{2}}}dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca0bf83a81a93a7672386b943ca4db8bb1b5489)
Wzory Eulera
![{\displaystyle e^{ix}=\cos x+i\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8)
![{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a19a293592d5d5dda850bf2de5b92aba3c9764f)
![{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e16aed66a59735cc85653fb255bd6ba5b3e0db4)
![{\displaystyle \operatorname {tg} x={\frac {e^{ix}-e^{-ix}}{(e^{ix}+e^{-ix})i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3005113d8be4806aed8587bfdce0bded71487c32)
![{\displaystyle \operatorname {ctg} x={\frac {e^{ix}+e^{-ix}}{e^{ix}-e^{-ix}}}i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4406efca438902b89a8eb7dc2fd17184f689ac)
Wzory te pozwalają łatwo przekształcać wyrażenia trygonometryczne, poprzez przejście na postać zespoloną (cztery ostatnie wzory), uproszczenie i powrót na postać trygonometryczną (pierwszy wzór).
Inne zależności między funkcjami trygonometrycznymi
![{\displaystyle \operatorname {tg} x+\sec x=\operatorname {tg} \left({\frac {x}{2}}+{\frac {\pi }{4}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc401c76cea7b364cc4f766627dddf58063f73a)
Wzór de Moivre’a
![{\displaystyle \cos nx+i\sin nx=(\cos x+i\sin x)^{n}\qquad n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b300766cb2878b8f6f86e8efded433f116df7895)
lub ogólniej:
![{\displaystyle [r(\cos x+i\sin x)]^{n}=r^{n}(\cos nx+i\sin nx)\qquad n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/76357c43423817aec6638c42c3dfecbcfbbcfde5)
Zobacz też
- trygonometryczne wzory redukcyjne