ຕຳລາໄຕມຸມ

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.
• $\tan(\alpha +\beta )={\frac {\tan \alpha +\tan \beta }{1-\tan \alpha \,\tan \beta }}.$
• $\tan(\alpha -\beta )={\frac {\tan \alpha -\tan \beta }{1+\tan \alpha \,\tan \beta }}.$

• sin 2α = 2 sin α cos α.
• cos 2α = cos² α − sin² α = 2 cos² α − 1 = 1 − 2 sin² α.
• $\tan 2\alpha ={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}$

• sin 3α = −4sin³ α + 3sin α.
• cos 3α = 4cos³ α − 3cos α.
• $\tan 3\alpha ={\frac {3\tan \alpha -\tan ^{3}\alpha }{1-3\tan ^{2}\alpha }}$

• $\sin ^{2}\!\left({\frac {\alpha }{2}}\right)={\frac {1-\cos \alpha }{2}}.$
• $\cos ^{2}\!\left({\frac {\alpha }{2}}\right)={\frac {1+\cos \alpha }{2}}.$
• $\sin \!\left({\frac {\alpha }{2}}\right)\cos \!\left({\frac {\alpha }{2}}\right)={\frac {\sin \alpha }{2}}.$

• $\sin ^{3}\alpha ={1 \over 4}(3\sin \alpha -\sin 3\alpha ).$
• $\cos ^{3}\alpha ={1 \over 4}(3\cos \alpha +\cos 3\alpha ).$

• $\sin \alpha +\sin \beta =2\sin \!\left({\alpha +\beta \over 2}\right)\cos \!\left({\alpha -\beta \over 2}\right).$
• $\sin \alpha -\sin \beta =2\cos \!\left({\alpha +\beta \over 2}\right)\sin \!\left({\alpha -\beta \over 2}\right).$
• $\cos \alpha +\cos \beta =2\cos \!\left({\alpha +\beta \over 2}\right)\cos \!\left({\alpha -\beta \over 2}\right).$
• $\cos \alpha -\cos \beta =-2\sin \!\left({\alpha +\beta \over 2}\right)\sin \!\left({\alpha -\beta \over 2}\right).$

• $\sin \alpha \,\cos \beta ={1 \over 2}\{\sin(\alpha +\beta )+\sin(\alpha -\beta )\}.$
• $\cos \alpha \,\sin \beta ={1 \over 2}\{\sin(\alpha +\beta )-\sin(\alpha -\beta )\}.$
• $\cos \alpha \,\cos \beta ={1 \over 2}\{\cos(\alpha +\beta )+\cos(\alpha -\beta )\}.$
• $\sin \alpha \,\sin \beta =-{1 \over 2}\left\{\cos(\alpha +\beta )-\cos(\alpha -\beta )\right\}.$

• $a\sin \theta +b\cos \theta ={\sqrt {a^{2}+b^{2}}}\sin(\theta +\phi ),$

ແຕ່ $\phi =\tan ^{-1}\!\left({\frac {b}{a}}\right).$

$\sin {({\pi }z)}={\pi }z\prod _{n=1}^{\infty }{\left(1-{\frac {z^{2}}{n^{2}}}\right)}$

$\cos {({\pi }z)}=\prod _{n=1}^{\infty }{\left(1-{\frac {z^{2}}{(n-{\frac {1}{2}})^{2}}}\right)}$

$\pi \cot {{\pi }z}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {2z}{z^{2}-n^{2}}}$

$\pi \tan {{\pi }z}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {-1}{z+\textstyle {\frac {1}{2}}+n}}=-\sum _{n=0}^{\infty }{\frac {2z}{z^{2}-\left(n+\textstyle {\frac {1}{2}}\right)^{2}}}$

${\frac {\pi }{\sin {\pi }z}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}2z}{z^{2}-n^{2}}}$

${\frac {\pi }{\cos {{\pi }z}}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+{\frac {1}{2}}+n}}=-\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)}{z^{2}-\left(n+{\frac {1}{2}}\right)^{2}}}$

• ${d \over dx}\sin x=\cos x,$
• ${d \over dx}\cos x=-\sin x,$
• ${d \over dx}\tan x=\sec ^{2}x=1+\tan ^{2}x.$

$x=\sin y\iff y=\sin ^{-1}x,$

$x=\cos y\iff y=\cos ^{-1}x,$

$x=\tan y\iff y=\tan ^{-1}x,$

$x=\cot y\iff y=\cot ^{-1}x,$

$x=\sec y\iff y=\sec ^{-1}x,$

$-{\frac {\pi }{2}}\leq \sin ^{-1}x\leq {\frac {\pi }{2}},$

$0\leq \cos ^{-1}x\leq \pi ,$

exp(ix) = cos x + i sin x

exp(−ix) = cos x − i sin x

$\cos x={\frac {e^{ix}+e^{-ix}}{2}},$

$\sin x={\frac {e^{ix}-e^{-ix}}{2i}}.$