ຕຳລາໄຕມຸມ

ຈາກ ວິກິພີເດຍ

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}\le\sin^{-1}x\le\frac{\pi}{2},
0\le\cos^{-1}x\le\pi,
exp(ix) = cos x + i sin x
exp(−ix) = cos xi sin x
\cos x = \frac{e^{ix}+e^{-ix}}{2},
\sin x = \frac{e^{ix}-e^{-ix}}{2i}.