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初等関数 (elementary function)

初等関数 (elementary function)

三角関数

tanx=sinxcosxcotx=cosxsinx=1tanxsecx=1cosxcscx=1sinx
\tan x &=& \frac{\sin x}{\cos x}

\cot x &=& \frac{\cos x}{\sin x} = \frac{1}{\tan x}

\sec x &=& \frac{1}{\cos x}

\csc x &=& \frac{1}{\sin x}

sin2x+cos2=11+tan2x=sec2x
\sin ^2 x + \cos ^2 &=& 1

1 + \tan ^2 x &=& \sec ^2 x

sin(x)=sinxcos(x)=cosxtan(x)=tanx
\sin (-x) &=& -\sin x

\cos (-x) &=& \cos x

\tan (-x) &=& -\tan x

sin(x+y)=sinxcosy+cosxsinysin(xy)=sinxcosycosxsinysin(x±y)=sinxcosy±cosxsiny
\sin (x+y) &=& \sin x \cos y + \cos x \sin y

\sin (x-y) &=& \sin x \cos y – \cos x \sin y

\sin (x \pm y) &=& \sin x \cos y \pm \cos x \sin y

cos(x+y)=cosxcosysinxsinycos(xy)=cosxcosy+sinxsinycos(x±y)=cosxcosysinxsiny
\cos (x+y) &=& \cos x \cos y – \sin x \sin y

\cos (x-y) &=& \cos x \cos y + \sin x \sin y

\cos (x \pm y) &=& \cos x \cos y \mp \sin x \sin y

tan(x+y)=tanx+tany1tanxtanytan(xy)=tanxtany1+tanxtanytan(x±y)=tanx±tany1tanxtany
\tan (x+y) &=& \frac{\tan x + \tan y}{1 – \tan x \tan y}

\tan (x-y) &=& \frac{\tan x – \tan y}{1 + \tan x \tan y}

\tan (x \pm y) &=& \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}

sinx+siny=2sinx+y2cosxy2sinxsiny=2cosx+y2sinxy2
\sin x + \sin y &=& 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}

\sin x – \sin y &=& 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}

cosx+cosy=2cosx+y2cosxy2cosxcosy=2sinx+y2sinxy2
\cos x + \cos y &=& 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}

\cos x – \cos y &=& -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}

sin2x2=12(1cosx)cos2x2=12(1+cosx)tan2x2=1cosx1+cosx
\sin ^2 \frac{x}{2} &=& \frac{1}{2} (1 – \cos x)

\cos ^2 \frac{x}{2} &=& \frac{1}{2} (1 + \cos x)

\tan ^2 \frac{x}{2} &=& \frac{1 – \cos x}{1 + \cos x}

sinxsiny=12{cos(x+y)cos(xy)}cosxcosy=12{cos(x+y)+cos(xy)}sinxcosy=12{sin(x+y)+sin(xy)}
\sin x \sin y &=& -\frac{1}{2} \{ \cos (x+y) – \cos (x-y) \}

\cos x \cos y &=& \frac{1}{2} \{ \cos (x+y) + \cos (x-y) \}

\sin x \cos y &=& \frac{1}{2} \{ \sin (x+y) + \sin (x-y) \}

指数・対数関数

指数の基本法則

am×an=am+na0=1(am)n=amn(ab)m=anbna1=1ana1n=na
a^m \times a^n &=& a^{m+n}

a^0 &=& 1

(a^m)^n &=& a^{mn}

(ab)^m &=& a^n \cdot b^n

a^{-1} &=& \frac{1}{a^n}

a^{\frac{1}{n}} &=& ^n\sqrt{a}

対数の基本法則

logaxy=logax+logaylogaxu=ulogaxlogax=logbxlogbaloga1=0
x=eylogex=logeeylogex=ylogeelogex=y
x &=& e^y

\log _e x &=& \log_e e^y

\log _e x &=& y \log_e e

\log _e x &=& y

x=10ylog10x=log1010ylog10x=ylog1010log10x=y
x &=& 10^y

\log _{10} x &=& \log_{10} 10^y

\log _{10} x &=& y \log_{10} 10

\log _{10} x &=& y

x=aylogax=logaaylogax=ylogaalogax=y
x &=& a^y

\log _a x &=& \log_a a^y

\log _a x &=& y \log_a a

\log _a x &=& y

logex=log10xlog10e
\log _e x = \frac{\log _{10} x}{\log _{10} e}
log(xy)=logx+logylogxy=logxlogylogxα=αlogxddxlogx=1x
\log (xy) &=& \log x + \log y

\log \frac{x}{y} &=& \log x – \log y

\log x^{\alpha} &=& \alpha \log x

\frac{d}{dx} \log x &=& \frac{1}{x}

双曲線正弦関数・双曲線余弦関数

sinhx=exex2coshx=ex+ex2
\sinh x &=& \frac{e^x – e^{-x}}{2}

\cosh x &=& \frac{e^x + e^{-x}}{2}

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