Rhestr unfathiannau trigonometrig

Oddi ar Wicipedia
Neidio i: llywio, chwilio

Nodiant[golygu]

Defnyddir y nodiant canlynol ar gyfer pob un o'r chwech ffwythiant trigonometrig (sin, cosin (cos), tangiad (tan), cotangiad (cot), secant (sec), a chosecant (csc). Dim ond y nodiant ar gyfer sin a roddir isod, mae'r nodiant ar gyfer y ffwythiannau eraill yn gyffelyb.

Nodiant Darllener Disgrifiad Diffiniad
sin²(x) "sin sgwâr x" sin wedi ei sgwario sin²(x) = (sin(x))²
arcsin(x) "arcsin x" ffwythiant gwrthdro sin arcsin(x) = y  os a dim ond os  sin(y) = x a -{\pi \over 2} \le y \le {\pi \over 2}
(sin(x))−1 "sin x, i'r [pŵer] meinws un" Cilydd sin (sin(x))−1 = 1 / sin(x) = csc(x)

Gellir ysgrifennu arcsin(x) yn sin−1(x) yn ogystal; rhaid gofalu rhag drysu hyn â (sin(x))−1.

Diffiniadau[golygu]

\begin{align}
\cos(x) &= \sin\left( x + \frac {\pi} {2}\right)\\
\tan(x) &= \frac{\sin(x)}{\cos(x)} &\quad \cot(x)&= \frac{\cos(x)}{\sin(x)} = \frac{1}{\tan(x)}\\
\sec(x) &= \frac{1}{\cos(x)} &\quad \csc(x)&= \frac{1}{\sin(x)}
\end{align}

(Gweler ffwythiant trigonometrig am fwy o wybodaeth)

Cyfnodedd, cymesuredd a symudiadau[golygu]

Cyfnodedd[golygu]

Mae cyfnod o 2π gan y ffwythiannau sin, cosin, secant, a chosecant (cylch llawn): Os y mae k yn unrhyw gyfanrif yna mae

\begin{align}
\sin(x) &= \sin(x + 2k\pi) \\
\cos(x) &= \cos(x + 2k\pi) \\
\sec(x) &= \sec(x + 2k\pi) \\
\csc(x) &= \csc(x + 2k\pi) \\
\end{align}

Mae cyfnod o π (hanner cylch) gan y ffwythiannau tangiad a chotangiad:

\begin{align}
\tan(x) &= \tan(x + k\pi) \\
\cot(x) &= \cot(x + k\pi) \\
\end{align}

Cymesuredd[golygu]


\begin{align}
\sin(-x) &= -\sin(x) & \sin\left(\tfrac{\pi}{2} - x\right) &= \cos(x) & \sin\left(\pi - x\right) &= +\sin(x)   \\
\cos(-x) &= +\cos(x) & \cos\left(\tfrac{\pi}{2} - x\right) &= \sin(x) & \cos\left(\pi - x\right) &= -\cos(x)      \\
\tan(-x) &= -\tan(x) & \tan\left(\tfrac{\pi}{2} - x\right) &= \cot(x) & \tan\left(\pi - x\right) &= -\tan(x)      \\
\cot(-x) &= -\cot(x) & \cot\left(\tfrac{\pi}{2} - x\right) &= \tan(x) & \cot\left(\pi - x\right) &= -\cot(x)      \\
\sec(-x) &= +\sec(x) & \sec\left(\tfrac{\pi}{2} - x\right) &= \csc(x) & \sec\left(\pi - x\right) &= -\sec(x)      \\
\csc(-x) &= -\csc(x) & \csc\left(\tfrac{\pi}{2} - x\right) &= \sec(x) & \csc\left(\pi - x\right) &= +\csc(x)
\end{align}

Symudiadau[golygu]


\begin{align}
\sin\left(x + \tfrac{\pi}{2}\right) &= +\cos(x) & \sin\left(x + \pi\right) &= -\sin(x)   \\
\cos\left(x + \tfrac{\pi}{2}\right) &= -\sin(x) & \cos\left(x + \pi\right) &= -\cos(x)   \\
\tan\left(x + \tfrac{\pi}{2}\right) &= -\cot(x) & \tan\left(x + \pi\right) &= +\tan(x)   \\
\cot\left(x + \tfrac{\pi}{2}\right) &= -\tan(x) & \cot\left(x + \pi\right) &= +\cot(x)   \\
\sec\left(x + \tfrac{\pi}{2}\right) &= -\csc(x) & \sec\left(x + \pi\right) &= -\sec(x)   \\
\csc\left(x + \tfrac{\pi}{2}\right) &= +\sec(x) & \csc\left(x + \pi\right) &= -\csc(x)
\end{align}

Cyfuniadau llinol[golygu]

Weithiau mae'n bwysig gwybod bod cyfuniad llinol o donau sin gyda'r un cyfnod (ond gyda gwahanol symudiad cydwedd) yn rhoi ton sin gyda'r un cyfnod. Yn gyffrefinol, mae

a\sin x+b\cos x=\sqrt{a^2+b^2}\cdot\sin(x+\varphi)\,

lle mae


  \varphi=
  \left\{
   \begin{matrix}
    {\rm arctan}(b/a),&&\mbox{os mae }a\ge0; \;
   \\
    \arctan(b/a) \pm \pi,&&\mbox{os mae }a<0. \;
   \end{matrix}
  \right. \;

Yn gyffredinol, am symudiad cydwedd mympwyol, mae gennym fod

a\sin x+b\sin(x+\alpha)= c \sin(x+\beta)\,

lle mae


  c = \sqrt{a^2 + b^2 +2ab\cos \alpha},

a


  \beta = {\rm arctan} \left(\frac{b\sin \alpha}{a + b\cos \alpha}\right).

Unfathiannau Pythagoreaidd[golygu]

Seilir y canlynol ar theorem Pythagoras:

\begin{align}
\sin^2(x) + \cos^2(x) &= 1 \\
        \tan^2(x) + 1 &= \sec^2(x) \\
        \cot^2(x) + 1 &= \csc^2(x)
\end{align}

Gellir deillio'r ail a'r trydydd hafaliad uchod o'r cyntaf trwy rhannu â cos2(x) a sin2(x) yn ol eu trefn.

Unfathiannau swm neu wahaniaeth onglau[golygu]

Fe'u celwir hefyd yn "fformwlâu adio a thynnu". Gellir eu profi gan ddefnyddio fformwla Euler.

\sin(x \pm y) = \sin(x) \cos(y) \pm \cos(x) \sin(y)\,
(Pan y mae "+" ar y chwith, mae "+" ar y de, ac yn gyffelyb gyda "-".)
\cos(x \pm y) = \cos(x) \cos(y) \mp \sin(x) \sin(y)\,
(Pan y mae "+" ar y chwith, mae "-" ar y de, ac i'r gwrthwyneb.)
\tan(x \pm y) = \frac{\tan(x) \pm \tan(y)}{1 \mp \tan(x)\tan(y)}

Tangiad symiau nifer meidraidd o dermau[golygu]

Gadewch i xi = tan(θi ), ar gyfer i = 1, ..., n. Gadewch i ek fod y polynomial cymesur elfennol gyda gradd k yn y newidynnau xi, i = 1, ..., n, k = 0, ..., n. Yna mae

\tan(\theta_1+\cdots+\theta_n) = \frac{e_1 - e_3 + e_5 -\cdots}{e_0 - e_2 + e_4 - \cdots},

gyda'r nifer o dermau yn dibynnu ar n.

Er enghraifft, mae

 \begin{align} \tan(\theta_1 + \theta_2 + \theta_3)
&{}= \frac{e_1 - e_3}{e_0 - e_2} = \frac{(x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3)}{
1 \ - \ (x_1 x_2 + x_1 x_3 + x_2 x_3)}, \\  \\
\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4)
&{}= \frac{e_1 - e_3}{e_0 - e_2 + e_4} \\  \\
&{}= \frac{(x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4)}{
1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4)},\end{align}

ac yn y blaen. Gellir profi hyn trwy anwythiad mathemategol.

Fformwlâu ongl dwbl[golygu]

Gellir profi'r canlynol trwy amnewid x = y yn y fformwlâu adio, a defnyddio'r fformwla Pythagoreaidd, neu trwy ddefnyddio fformwla de Moivre gydag n = 2.

\sin(2x) = 2 \sin (x) \cos(x) \,
\cos(2x) = \cos^2(x) - \sin^2(x)
  = 2 \cos^2(x) - 1 = 1 - 2 \sin^2(x)  = \frac{1 - \tan^2(x)} {1 + \tan^2(x)} \,
 \tan(2x) = \frac{2 \tan (x)} {1 - \tan^2(x)}\,


\cot(2x) = \frac{\cot(x) - \tan(x)}{2}\,

Gellir defnyddio'r uchod i ganfod triawdau Pythagoreaidd. Os y mae (a, b, c) yw hyd ochrau triongl ongl-sgwâr, yna mae (a2 − b2, 2ab, c2) hefyd yn ffurfio triongl ongl-sgwâr, lle mae B yw'r ongl a ddyblir. Os y mae a2 − b2 yn negatif, cymerwch ei wrthdro a defnyddio ongl cyflenwol 2B yn lle 2B.

Fformwlâu ongl triphlyg[golygu]

\sin(3x)= 3 \sin(x)- 4 \sin^3(x) \,
\cos(3x)= 4 \cos^3(x) - 3 \cos(x) \,
\tan(3x)= \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2(x)}

Fformwlâu aml-ongl[golygu]

Os mai Tn yw'r nfed polynomial Chebyshev, yna mae

\cos(nx)=T_n(\cos(x)). \,

Os mai Sn yw'r nfed polynomial gwasgar, yna mae

\sin^2(n\theta) = S_n(\sin^2\theta).\,

Fformwla de Moivre:

\cos(nx)+i\sin(nx)=(\cos(x)+i\sin(x))^n \,

Fformwlâu lleihau pŵer[golygu]

\sin^2(x) = {1 - \cos(2x) \over 2}
\cos^2(x) = {1 + \cos(2x) \over 2}
\sin^2(x) \cos^2(x) = {1 - \cos(4 x) \over 8}
\sin^3(x) = \frac{3 \sin(x) - \sin(3 x)}{4}
\cos^3(x) = \frac{3 \cos(x) + \cos(3 x)}{4}

Fformwlâu hanner ongl[golygu]

\cos\left(\frac{x}{2}\right) =  \pm\, \sqrt{\frac{1 + \cos(x)}{2}}
\sin\left(\frac{x}{2}\right) =  \pm\, \sqrt{\frac{1 - \cos(x)}{2}}


 \tan\left(\frac{x}{2}\right) = {\sin (x/2) \over \cos (x/2)} = \pm\, \sqrt{1 - \cos x \over 1 + \cos x}. \qquad \qquad (1)


 \tan\left(\frac{x}{2}\right) = \pm\, \sqrt{(1 - \cos x) (1 + \cos x) \over (1 + \cos x) (1 + \cos x)} = \pm\, \sqrt{1 - \cos^2 x \over (1 + \cos x)^2}
 = {\sin x \over 1 + \cos x}.


 \tan\left(\frac{x}{2}\right) = \pm\, \sqrt{(1 - \cos x) (1 - \cos x) \over (1 + \cos x) (1 - \cos x)} = \pm\, \sqrt{(1 - \cos x)^2 \over (1 - \cos^2 x)}
 = {1 - \cos x \over \sin x}.


\tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)} = \frac{1-\cos(x)}{\sin(x)}.


\tan\left({x \over 2}\right) = \csc(x) - \cot(x),
\cot\left({x \over 2}\right) = \csc(x) + \cot(x).


t = \tan\left(\frac{x}{2}\right),


    \sin(x) = \frac{2t}{1 + t^2}   a   \cos(x) = \frac{1 - t^2}{1 + t^2}   a   e^{i x} = \frac{1 + i t}{1 - i t}.

Amnewidiad o t am tan(x/2) yw hyn, gyda'r canlyniad fod sin(x) yn newid yn 2t/(1 + t2) a cos(x) yn (1 − t2)/(1 + t2). Mae hyn yn ddefnyddiol mewn calcwlws ar gyfer integreiddio ffwythiannau cymarebol o sin(x) a cos(x).

Unfathiannau lluoswm-i-swm[golygu]

\cos\left (x\right ) \cos\left (y\right ) = {\cos\left (x - y\right ) + \cos\left (x + y\right ) \over 2} \;
\sin\left (x\right ) \sin\left (y\right ) = {\cos\left (x - y\right ) - \cos\left (x + y\right ) \over 2} \;
\sin\left (x\right ) \cos\left (y\right ) = {\sin\left (x - y\right ) + \sin\left (x + y\right ) \over 2} \;

(gw. Theorem Ptolemy)

Unfathiannau swm-i-lluoswm[golygu]

\cos(x) + \cos(y) = 2 \cos\left( \frac{x + y}{2} \right) \cos\left( \frac{x - y}{2} \right) \;
\sin(x) + \sin(y) = 2 \sin\left( \frac{x + y}{2} \right) \cos\left( \frac{x - y}{2} \right) \;
 \cos(x) - \cos(y) = -2 \sin\left( {x + y \over 2}\right) \sin\left({x - y \over 2}\right) \;
 \sin(x) - \sin(y) = 2 \cos\left({x + y\over 2}\right) \sin\left({x - y\over 2}\right) \;

fformwla de Moivre

\mbox{os mae }x + y + z = \pi ,
\mbox{yna mae }\tan(x) + \tan(y) + \tan(z) = \tan(x)\tan(y)\tan(z).\,

(Os am roi ystyr i'r fformwla tra fod unrhyw un o x, y, a z yn ongl sgwâr, rhaid cymryd mai ∞ yw'r ddau ochr. Nid +∞ neu −∞ yw hyn, ond un pwynt "at anfeidredd" a ychwanegir i'r linell rif real.)

\mbox{Os mae }x + y + z = \pi = \mbox{hanner cylch,}\,
\mbox{yna mae }\sin(2x) + \sin(2y) + \sin(2z) = 4\sin(x)\sin(y)\sin(z).\,

Ffwythiannau trigonometrig gwrthdro[golygu]

 \arcsin(x)+\arccos(x)=\pi/2\;
 \arctan(x)+\arccot(x)=\pi/2.\;
\arctan(x)+\arctan(1/x)=\left\{\begin{matrix} \pi/2, & \mbox{Os mae }x > 0 \\  -\pi/2, & \mbox{os mae }x < 0 \end{matrix}\right.
\arctan(x)+\arctan(y)=\arctan\left(\frac{x+y}{1-xy}\right)+\left\{\begin{matrix} \pi, & \mbox{os mae }x,y>0 \\ -\pi, & \mbox{os mae }x,y<0 \\ 0, & \mbox{fel arall } \end{matrix}\right.
\sin[\arccos(x)]=\sqrt{1-x^2} \,
\cos[\arcsin(x)]=\sqrt{1-x^2} \,
\sin[\arctan(x)]=\frac{x}{\sqrt{1+x^2}}
\cos[\arctan(x)]=\frac{1}{\sqrt{1+x^2}}
\tan[\arcsin (x)]=\frac{x}{\sqrt{1 - x^2}}
\tan[\arccos (x)]=\frac{\sqrt{1 - x^2}}{x}

Perthynas gyda'r ffwythiant esbonyddol cymhlyg[golygu]

e^{ix} = \cos(x) + i\sin(x)\,
e^{-ix} = \cos(x) - i\sin(x)\,
\cos(x) = \frac{e^{ix} + e^{-ix}}{2} \;
\sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \;

lle mae i 2 = −1.

Gw. fformwla Euler.

Diffiniadau esbonyddol[golygu]

\sin (\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \,
\cos (\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \,
\tan (\theta) = \frac{\sin (\theta)}{\operatorname{cosh} (\theta)} = \frac{(\frac{e^{i\theta} - e^{-i\theta}}{2i})}{(\frac{e^{i\theta} + e^{-i\theta}}{2})} \,
\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)} = \frac{(\frac{e^{i\theta} + e^{-i\theta}}{2})}{(\frac{e^{i\theta} - e^{-i\theta}}{2i})} \,
\sec (\theta) = \frac{1}{\cos (\theta)} = \frac{1}{(\frac{e^{i\theta} + e^{-i\theta}}{2})} \,
\csc (\theta) = \frac{1}{\sin (\theta)} = \frac{1}{(\frac{e^{i\theta} - e^{-i\theta}}{2i})} \,
\operatorname{versin} (\theta) = 1 - \cos (\theta) = 1 - \frac{e^{i\theta} + e^{-i\theta}}{2} \,
\operatorname{vercos} (\theta) = 1 - \sin (\theta) = 1 - \frac{e^{i\theta} - e^{-i\theta}}{2i} \,
\operatorname{exsec} (\theta) = \operatorname{sec} (\theta) - 1 \ = \frac{1}{\cos (\theta)} - 1 = \frac{1}{(\frac{e^{i\theta} + e^{-i\theta}}{2})} - 1 \,
\operatorname{excsc} (\theta) = \operatorname{csc} (\theta) - 1 \ = \frac{1}{\sin (\theta)} - 1 = \frac{1}{(\frac{e^{i\theta} - e^{-i\theta}}{2i})} - 1 \,
\operatorname{sinh} (\theta) = -i\sin (i\theta) = \frac{e^{\theta} - e^{-\theta}}{2} \,
\operatorname{cosh} (\theta) = \cos (i\theta) = \frac{e^{\theta} + e^{-\theta}}{2} \,
\operatorname{tanh} (\theta) = -i\tan (i\theta) = \frac{\operatorname{sinh} (\theta)}{\operatorname{cosh} (\theta)} = \frac{e^\theta - e^{-\theta}}{e^\theta + e^{-\theta}} = \frac{e^{2\theta} - 1}{e^{2\theta} + 1} \,
\operatorname{coth} (\theta) = i\operatorname{cot} (i\theta) = \frac{\operatorname{cosh} (\theta)}{\operatorname{sinh} (\theta)} = \frac{e^\theta + e^{-\theta}}{e^\theta - e^{-\theta}} = \frac{e^{2\theta} + 1}{e^{2\theta} - 1} \,
\operatorname{sech} (\theta) = \frac{1}{\operatorname{cosh} (\theta)} = \operatorname{sec} (i\theta) = \frac{2}{e^{\theta} + e^{-\theta}} \,
\operatorname{csch} (\theta) = \frac{1}{\operatorname{sinh} (\theta)} = i \cos (i\theta) = \frac{2}{e^{\theta} - e^{-\theta}} \,
\operatorname{versinh} (\theta) = 1 - \cos (i\theta) = 1 - \frac{e^{\theta} + e^{-\theta}}{2} \,
\operatorname{vercosh} (\theta) = 1 + i\sin (i\theta) = 1 - \frac{e^{\theta} - e^{-\theta}}{2} \,
\operatorname{exsech} (\theta) = \operatorname{sech} (\theta) - 1 = \frac{1}{\operatorname{cosh} (\theta)} - 1 = \operatorname{sec} (i\theta) = \frac{2}{e^{\theta} + e^{-\theta}} - 1 \,
\operatorname{excsch} (\theta) = \operatorname{csch} (\theta) - 1 = \frac{1}{\operatorname{sinh} (\theta)} - 1 = i \cos (i\theta) = \frac{2}{e^{\theta} - e^{-\theta}} - 1 \,
\arcsin (\theta) = -i \ln (i\theta + \sqrt{1 - \theta^2}) \,
\arccos (\theta) = -i \ln (\theta + i\sqrt{1 - \theta^2}) \,
\arctan (\theta) = \frac{\ln (\frac{i + \theta}{i - \theta}) i}{2} \,
\arccot (\theta) = \arctan (-\theta) = \frac{i \ln (\frac{i - \theta}{i + 
\theta})}{2} \,
\arcsec (\theta) = \arccos (\theta^{-1}) = -i \ln (\theta^{-1} + \sqrt{1 - \theta^{-2}}i) \,
\arccsc (\theta) = \arcsin (\theta^{-1}) = -i \ln (i\theta^{-1} + \sqrt{1 - \theta^{-2}}) \,
\operatorname{arcversin} (\theta) = \arccos (1 - \theta) = -i \ln (1 - \theta + i\sqrt{1 - (1 - \theta)^2}) \,
\operatorname{arcvercos} (\theta) = \operatorname{arcsin} (1 - \theta) = -i \ln (i - i\theta + \sqrt{1 - (1 - \theta)^2}) \,
\operatorname{arcexsec} (\theta) = \arcsec (1 + \theta) = -i \ln ((\theta + 1)^{-1} + i \sqrt{1 - (1 + \theta)^2}) \,
\operatorname{arcexcsc} (\theta) = \arccsc (1 + \theta) = -i \ln (i (\theta + 1)^{-1} + \sqrt{1 - (1 + \theta)^2}) \,
\operatorname{arcsinh} (\theta) = \ln (\theta + \sqrt{\theta^2 + 1}) \,
\operatorname{arccosh} (\theta) = \ln (\theta + \sqrt{\theta^2 - 1}) \,
\operatorname{arctanh} (\theta) = \frac{\ln (\frac{i + \theta}{i - \theta})}{2} \,
\operatorname{arccoth} (\theta) = \operatorname{arctanh} (-\theta) = \frac{\ln (\frac{i - \theta}{i + \theta})}{2} \,
\operatorname{arcsech} (\theta) = \operatorname{arccosh} (\theta^{-1}) = \ln (\theta^{-1} + \sqrt{\theta^{-2} - 1}) \,
\operatorname{arccsch} (\theta) = \operatorname{arcsinh} (\theta^{-1}) = \ln (\theta^{-1} + \sqrt{\theta^{-2} + 1}) \,
\operatorname{arcversinh} (\theta) = \operatorname{arccosh} (\theta) - 1 = \ln (\theta + \sqrt{\theta^2 - 1}) - 1 \,
\operatorname{arcvercosh} (\theta) = \operatorname{arcsinh} (\theta) - 1 = \ln (\theta + \sqrt{\theta^2 + 1}) - 1\,
\operatorname{arcexsech} (\theta) = \operatorname{arcsech} (\theta + 1) = \operatorname{arccosh} ((\theta + 1)^{-1}) = \ln ((\theta + 1)^{-1} + \sqrt{(\theta + 1)^{-2} - 1}) \,
\operatorname{arcexcsch} (\theta) = \operatorname{arccsch} (\theta + 1) = \operatorname{arcsinh} ((\theta + 1)^{-1}) = \ln ((\theta + 1)^{-1} + \sqrt{(\theta + 1)^{-2} + 1}) \,