Rhestr unfathiannau trigonometrig

## Nodiant
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.

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

## Diffiniadau 
:\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 
### Cyfnodedd 
Mae cyfnod o 2π gan y ffwythiannau sin, cosin, secant, a chosecant (cylch llawn): os 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 
:
\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 
:
\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 
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

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 
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 ôl eu trefn.

## Unfathiannau swm neu wahaniaeth onglau 
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 
Gadewch i *x**i* = tan(θ*i*&nbsp;), ar gyfer *i* = 1,&nbsp;...,&nbsp;*n*. Gadewch i *e**k* fod y polynomial cymesur elfennol gyda gradd *k* yn y newidynnau *x**i*, *i*&nbsp;=&nbsp;1,&nbsp;...,&nbsp;*n*, *k*&nbsp;=&nbsp;0,&nbsp;...,&nbsp;*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 
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 Pythagoraidd. os mae (*a*, *b*, *c*) yw hyd ochrau triongl ongl-sgwâr, yna mae (*a*2&nbsp;−&nbsp;*b*2,&nbsp;2*ab*, *c*2) hefyd yn ffurfio triongl ongl-sgwâr, lle mae *B* yw'r ongl a ddyblir. os mae *a*2&nbsp;−&nbsp;*b*2 yn negatif, cymerwch ei wrthdro a defnyddio ongl cyflenwol 2*B* yn lle 2*B*.

## Fformwlâu ongl triphlyg 
:\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 
Os mai *Tn* yw'r *n*fed polynomial Chebyshev, yna mae
:\cos(nx)=T_n(\cos(x)). \,

Os mai *S**n* yw'r *n*fed 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 
: \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 
:\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),

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

## Unfathiannau lluoswm-i-swm 
:\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 Ptolemi)

## Unfathiannau swm-i-lluoswm 
: \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 
: \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 

:\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

:\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 
: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*&nbsp;2 = −1.

Gw. fformwla Euler.

## Diffiniadau esbonyddol 
: \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}) \,

Categori:Trigonometreg
unfathiannau trigonometrig