Tożsamości trygonometryczne – podstawowe zależności pomiędzy funkcjami trygonometrycznymi.
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)
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[edytuj | edytuj kod]
![{\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[edytuj | edytuj kod]
![{\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[edytuj | edytuj kod]
![{\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[edytuj | edytuj kod]
![{\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[edytuj | edytuj kod]
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.
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[edytuj | edytuj kod]
![{\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[edytuj | edytuj kod]
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)
![{\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)
![{\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)
![{\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)
![{\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[edytuj | edytuj kod]
![{\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)
Osobny artykuł: Wzór 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[edytuj | edytuj kod]
![{\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)