ຕຳລາໄຕມຸມ(ພາສາອັງກິດ:trigonometric function)ແມ່ນ ບັນດາຕຳລາຄະນິດສາດ ທີ່ເກີດມາຈາກ ທິດສະດີຮູບສາມແຈ.
- ຄວາມສຳພັນພື້ນຖານ
- sin(α + β) = sin α cos β + cos α sin β.
- sin(α − β) = sin α cos β − cos α sin β.
- cos(α + β) = cos α cos β − sin α sin β.
- cos(α − β) = cos α cos β + sin α sin β.
![{\displaystyle \tan(\alpha +\beta )={\frac {\tan \alpha +\tan \beta }{1-\tan \alpha \,\tan \beta }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c20ef37e44645c5e1ef7e30f711accef0c3c093)
![{\displaystyle \tan(\alpha -\beta )={\frac {\tan \alpha -\tan \beta }{1+\tan \alpha \,\tan \beta }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3af7b8ac309ce44b1b9fcec0a44bd2be6d7a746b)
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- sin 2α = 2 sin α cos α.
- cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α.
![{\displaystyle \tan 2\alpha ={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ea23121a61a4dfdd4f0768c6b1b2ca549b2d4a)
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- sin 3α = −4sin3 α + 3sin α.
- cos 3α = 4cos3 α − 3cos α.
![{\displaystyle \tan 3\alpha ={\frac {3\tan \alpha -\tan ^{3}\alpha }{1-3\tan ^{2}\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af49761b759ca94fb9540c04a46d0f70aa1f9929)
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![{\displaystyle \sin ^{2}\!\left({\frac {\alpha }{2}}\right)={\frac {1-\cos \alpha }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af73284e338e4ecb3c68d1322b16f05fcfcfa16e)
![{\displaystyle \cos ^{2}\!\left({\frac {\alpha }{2}}\right)={\frac {1+\cos \alpha }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e202edb98585a328f99f2154a648ed0e25d89f2)
![{\displaystyle \sin \!\left({\frac {\alpha }{2}}\right)\cos \!\left({\frac {\alpha }{2}}\right)={\frac {\sin \alpha }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bab02a4aae2da6e460bcbdde07758d06d4176165)
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![{\displaystyle \sin ^{3}\alpha ={1 \over 4}(3\sin \alpha -\sin 3\alpha ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03f8800ee35ec143d72e4e10f211cc83cd7ea2fe)
![{\displaystyle \cos ^{3}\alpha ={1 \over 4}(3\cos \alpha +\cos 3\alpha ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d7952fded7db55b410f8faf4deaba2d3a6e3e4)
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![{\displaystyle \sin \alpha +\sin \beta =2\sin \!\left({\alpha +\beta \over 2}\right)\cos \!\left({\alpha -\beta \over 2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f47f3bd52b2df3fe3dbabd0dd21215f5cb49d8c)
![{\displaystyle \sin \alpha -\sin \beta =2\cos \!\left({\alpha +\beta \over 2}\right)\sin \!\left({\alpha -\beta \over 2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0705b0489fa3ff3490ca25a6742b5fb0957aa9a)
![{\displaystyle \cos \alpha +\cos \beta =2\cos \!\left({\alpha +\beta \over 2}\right)\cos \!\left({\alpha -\beta \over 2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e40a38b07f5d7d7bf7a73e6df56b933d8275b6)
![{\displaystyle \cos \alpha -\cos \beta =-2\sin \!\left({\alpha +\beta \over 2}\right)\sin \!\left({\alpha -\beta \over 2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/245994166f31a118b986793b038924a6cf446e08)
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![{\displaystyle \sin \alpha \,\cos \beta ={1 \over 2}\{\sin(\alpha +\beta )+\sin(\alpha -\beta )\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94a9c54c35b1b3e87289f8dd8bff149d90bed425)
![{\displaystyle \cos \alpha \,\sin \beta ={1 \over 2}\{\sin(\alpha +\beta )-\sin(\alpha -\beta )\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cda510c0869d023935124b3073548a943a7e5837)
![{\displaystyle \cos \alpha \,\cos \beta ={1 \over 2}\{\cos(\alpha +\beta )+\cos(\alpha -\beta )\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/000d345000fa9733602edb4ffcf858f5e6b9ef14)
![{\displaystyle \sin \alpha \,\sin \beta =-{1 \over 2}\left\{\cos(\alpha +\beta )-\cos(\alpha -\beta )\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982264fcfe65d0b144f1bff7c92fc98e9522758a)
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![{\displaystyle a\sin \theta +b\cos \theta ={\sqrt {a^{2}+b^{2}}}\sin(\theta +\phi ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3b52824a5cd08690e497123e791df218717cec)
- ແຕ່
![{\displaystyle \phi =\tan ^{-1}\!\left({\frac {b}{a}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4227272c5b05ba5774aee66337c0f18f14632611)
![{\displaystyle \sin {({\pi }z)}={\pi }z\prod _{n=1}^{\infty }{\left(1-{\frac {z^{2}}{n^{2}}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/959d4b544fad7b5581a58c1d2658998cecff36f7)
![{\displaystyle \cos {({\pi }z)}=\prod _{n=1}^{\infty }{\left(1-{\frac {z^{2}}{(n-{\frac {1}{2}})^{2}}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccea01ff44fe91fd8e393d7209e814c44d8cc0b5)
![{\displaystyle \pi \cot {{\pi }z}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {2z}{z^{2}-n^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64e1d648dd2eb707033e9d125a0347439937352f)
![{\displaystyle \pi \tan {{\pi }z}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {-1}{z+\textstyle {\frac {1}{2}}+n}}=-\sum _{n=0}^{\infty }{\frac {2z}{z^{2}-\left(n+\textstyle {\frac {1}{2}}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26450b0bfc94ebe0ebccf6fa60d095b5547392bd)
![{\displaystyle {\frac {\pi }{\sin {\pi }z}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}2z}{z^{2}-n^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdfc57f5865ef0faf892d3c8f5458f21acbd1f2)
![{\displaystyle {\frac {\pi }{\cos {{\pi }z}}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+{\frac {1}{2}}+n}}=-\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)}{z^{2}-\left(n+{\frac {1}{2}}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/524ac11fc5f1b0e7d669580af6630570c618b478)
![{\displaystyle {d \over dx}\sin x=\cos x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/622836caf0d4b3fa0f842d7826e695c9b5b1d117)
![{\displaystyle {d \over dx}\cos x=-\sin x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aab81236476eb24406d03925661f84ebdb92e67)
![{\displaystyle {d \over dx}\tan x=\sec ^{2}x=1+\tan ^{2}x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/049d0fa3ddd5d6b31cf57ad10d92335f621755b6)
![{\displaystyle x=\sin y\iff y=\sin ^{-1}x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a653965e3b5920c1b8dd44baa0f74089d8404ef7)
![{\displaystyle x=\cos y\iff y=\cos ^{-1}x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87c4ae967f30d0d7e8013277e29fc461f1b688a2)
![{\displaystyle x=\tan y\iff y=\tan ^{-1}x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54bdce1ce6771c8135e9584458ca25cc3023345f)
![{\displaystyle x=\cot y\iff y=\cot ^{-1}x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23d27857a9c795b6ce96bc555c44ba397339b135)
![{\displaystyle x=\sec y\iff y=\sec ^{-1}x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1cd623d0fcfce5557e772f07bec37a94fefedf)
![{\displaystyle -{\frac {\pi }{2}}\leq \sin ^{-1}x\leq {\frac {\pi }{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd20af33bb5a0bfecc08155ebe3f99972e134840)
![{\displaystyle 0\leq \cos ^{-1}x\leq \pi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d01e9342646a05ab608d3a4145bc28e87f9f64fb)
- exp(ix) = cos x + i sin x
- exp(−ix) = cos x − i sin x
![{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50ddff1f9106e6aaa7032fcab7182e1022d517ea)
![{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646f51de572659a37a994a619ddcd871a20a1f47)