sin
x
=
e
i
x
−
e
−
i
x
2
i
cos
x
=
e
i
x
+
e
−
i
x
2
tan
x
=
e
i
x
−
e
−
i
x
(
e
i
x
+
e
−
i
x
)
i
arcsin
x
=
−
i
ln
(
1
−
x
2
+
i
x
)
arccos
x
=
−
i
ln
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x
2
−
1
+
x
)
arctan
x
=
−
i
2
ln
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1
+
i
x
1
−
i
x
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sh
x
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e
x
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e
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x
2
ch
x
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e
x
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e
−
x
2
th
x
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e
x
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e
−
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e
x
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e
−
x
arsh
x
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ln
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x
2
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1
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arch
x
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ln
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x
2
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1
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x
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arth
x
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1
2
ln
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1
+
x
1
−
x
)
\begin{aligned} \sin x &= \frac{e^{ix}-e^{-ix}}{2i} \\ \cos x &= \frac{e^{ix}+e^{-ix}}{2} \\ \tan x &= \frac{e^{ix}-e^{-ix}}{(e^{ix}+e^{-ix})i} \\ \arcsin x &= -i\ln(\sqrt{1-x^2} + ix) \\ \arccos x &= -i\ln(\sqrt{x^2-1} + x) \\ \arctan x &= -\frac i2\ln(\frac{1+ix}{1-ix}) \\ \sh x &= \frac{e^x-e^{-x}}{2} \\ \ch x &= \frac{e^x+e^{-x}}{2} \\ \th x &= \frac{e^x-e^{-x}}{e^x+e^{-x}} \\ \text{arsh }x &= \ln(\sqrt{x^2+1}+x) \\ \text{arch }x &= \ln(\sqrt{x^2-1}+x) \\ \text{arth }x &= \frac 12\ln(\frac{1+x}{1-x}) \end{aligned}
sinxcosxtanxarcsinxarccosxarctanxshxchxthxarsh xarch xarth x=2ieix−e−ix=2eix+e−ix=(eix+e−ix)ieix−e−ix=−iln(1−x2+ix)=−iln(x2−1+x)=−2iln(1−ix1+ix)=2ex−e−x=2ex+e−x=ex+e−xex−e−x=ln(x2+1+x)=ln(x2−1+x)=21ln(1−x1+x)
关系
sin
i
x
=
i
sh
x
sh
i
x
=
i
sin
x
cos
i
x
=
ch
x
ch
i
x
=
cos
x
tan
i
x
=
i
th
x
th
i
x
=
i
tan
x
arcsin
i
x
=
i
arsh
x
arsh
i
x
=
i
arcsin
x
arccos
i
x
=
−
i
arch
i
x
arch
i
x
=
i
arccos
i
x
arctan
i
x
=
i
arth
x
arth
i
x
=
i
arctan
x
\begin{aligned} \sin ix &= i\sh x \\ \sh ix &= i\sin x \\ \cos ix &= \ch x \\ \ch ix &= \cos x \\ \tan ix &= i\th x \\ \th ix &= i\tan x \\ \arcsin ix &= i \text{ arsh } x \\ \text{arsh } ix &= i\arcsin x \\ \arccos ix &= -i \text{ arch }ix \\ \text{arch } ix &= i \arccos ix \\ \arctan ix &= i\text{ arth } x \\ \text{arth } ix &= i\arctan x \end{aligned}
sinixshixcosixchixtanixthixarcsinixarsh ixarccosixarch ixarctanixarth ix=ishx=isinx=chx=cosx=ithx=itanx=i arsh x=iarcsinx=−i arch ix=iarccosix=i arth x=iarctanx
导数
(
sin
x
)
′
=
cos
x
(
cos
x
)
′
=
−
sin
x
(
tan
x
)
′
=
1
cos
2
x
(
arcsin
x
)
′
=
1
1
−
x
2
(
arccos
x
)
′
=
−
1
1
−
x
2
(
arctan
x
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′
=
1
1
+
x
2
(
sh
x
)
′
=
ch
x
(
ch
x
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′
=
sh
x
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th
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1
ch
2
x
(
arsh
x
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1
x
2
+
1
(
arch
x
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′
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1
x
2
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1
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arth
x
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′
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1
1
−
x
2
\begin{aligned} \left(\sin x\right)' &= \cos x \\ \left(\cos x\right)' &= -\sin x \\ \left(\tan x\right)' &= \frac{1}{\cos^2 x} \\ \left(\arcsin x\right)' &= \frac{1}{\sqrt{1-x^2}} \\ \left(\arccos x\right)' &= -\frac{1}{\sqrt{1-x^2}} \\ \left(\arctan x\right)' &= \frac{1}{1+x^2} \\ \left(\sh x\right)' &= \ch x \\ \left(\ch x\right)' &= \sh x \\ \left(\th x\right)' &= \frac{1}{\ch^2 x} \\ \left(\text{arsh } x\right)' &= \frac{1}{\sqrt{x^2+1}} \\ \left(\text{arch } x\right)' &= \frac{1}{\sqrt{x^2-1}} \\ \left(\text{arth } x\right)' &= \frac{1}{1-x^2} \\ \end{aligned}
(sinx)′(cosx)′(tanx)′(arcsinx)′(arccosx)′(arctanx)′(shx)′(chx)′(thx)′(arsh x)′(arch x)′(arth x)′=cosx=−sinx=cos2x1=1−x21=−1−x21=1+x21=chx=shx=ch2x1=x2+11=x2−11=1−x21
不定积分
∫
sin
x
d
x
=
−
cos
x
+
C
∫
cos
x
d
x
=
sin
x
+
C
∫
tan
x
d
x
=
−
ln
∣
cos
x
∣
+
C
∫
sh
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d
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ch
x
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C
∫
ch
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d
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sh
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C
∫
th
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x
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ln
∣
ch
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+
C
\begin{aligned} \int{\sin x\text{d}x} &= -\cos x+C \\ \int{\cos x\text{d}x} &= \sin x+C \\ \int{\tan x\text{d}x} &= -\ln\left|\cos x\right|+C \\ \int{\sh x\text{d}x} &= \ch x+C \\ \int{\ch x\text{d}x} &= \sh x+C \\ \int{\th x\text{d}x} &= \ln\left|\ch x\right|+C \\ \end{aligned}
∫sinxdx∫cosxdx∫tanxdx∫shxdx∫chxdx∫thxdx=−cosx+C=sinx+C=−ln∣cosx∣+C=chx+C=shx+C=ln∣chx∣+C
注:
arch
x
\text{arch } x
arch x 完整解析式为
±
ln
(
x
2
−
1
+
1
)
\pm\ln(\sqrt{x^2-1}+1)
±ln(x2−1+1), 且
arccos
x
=
±
i
arch
x
\arccos x = \pm i\text{ arch }x
arccosx=±i arch x.
