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微積分學/三角函數表
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編輯
<
微积分学
← 附錄
微積分學
求和符號 →
三角函數表
目次
1
定義
2
反三角函數
3
畢氏定理
4
倍角公式
5
和角公式
6
積化和差
7
復指數形式
8
雙曲函數
8.1
反雙曲函數
定義
編輯
tan
(
x
)
=
sin
(
x
)
cos
(
x
)
{\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}}
sec
(
x
)
=
1
cos
(
x
)
{\displaystyle \sec(x)={\frac {1}{\cos(x)}}}
cot
(
x
)
=
cos
(
x
)
sin
(
x
)
=
1
tan
(
x
)
{\displaystyle \cot(x)={\frac {\cos(x)}{\sin(x)}}={\frac {1}{\tan(x)}}}
csc
(
x
)
=
1
sin
(
x
)
{\displaystyle \csc(x)={\frac {1}{\sin(x)}}}
反三角函數
編輯
arcsin
(
x
)
=
∫
0
x
1
1
−
t
2
d
t
=
−
i
log
(
i
x
+
1
−
x
2
)
{\displaystyle \arcsin(x)=\int _{0}^{x}{\frac {1}{\sqrt {1-t^{2}}}}\mathrm {d} t=-i\log(ix+{\sqrt {1-x^{2}}})}
arccos
(
x
)
=
π
2
−
arcsin
(
x
)
=
π
2
−
∫
0
x
1
1
−
t
2
d
t
=
π
2
+
i
log
(
i
x
+
1
−
x
2
)
{\displaystyle \arccos(x)={\frac {\pi }{2}}-\arcsin(x)={\frac {\pi }{2}}-\int _{0}^{x}{\frac {1}{\sqrt {1-t^{2}}}}\mathrm {d} t={\frac {\pi }{2}}+i\log(ix+{\sqrt {1-x^{2}}})}
arctan
(
x
)
=
∫
0
x
1
1
+
t
2
d
t
=
i
2
log
(
1
−
i
x
1
+
i
x
)
{\displaystyle \arctan(x)=\int _{0}^{x}{\frac {1}{1+t^{2}}}\mathrm {d} t={\frac {i}{2}}\log \left({\frac {1-ix}{1+ix}}\right)}
arccsc
(
x
)
=
arcsin
(
1
x
)
=
−
i
log
(
i
x
+
1
−
1
z
2
)
{\displaystyle \operatorname {arccsc}(x)=\arcsin \left({\frac {1}{x}}\right)=-i\log \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{z^{2}}}}}\right)}
arcsec
(
x
)
=
arccos
(
1
x
)
=
π
2
−
arcsin
(
1
x
)
=
π
2
+
i
log
(
i
x
+
1
−
1
z
2
)
{\displaystyle \operatorname {arcsec}(x)=\arccos \left({\frac {1}{x}}\right)={\frac {\pi }{2}}-\arcsin \left({\frac {1}{x}}\right)={\frac {\pi }{2}}+i\log \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{z^{2}}}}}\right)}
arccot
(
x
)
=
arctan
(
1
x
)
=
π
2
−
arctan
(
x
)
=
π
2
+
i
2
log
(
1
+
i
x
1
−
i
x
)
{\displaystyle \operatorname {arccot}(x)=\arctan \left({\frac {1}{x}}\right)={\frac {\pi }{2}}-\arctan(x)={\frac {\pi }{2}}+{\frac {i}{2}}\log \left({\frac {1+ix}{1-ix}}\right)}
arcsin
(
x
)
+
arcsin
(
y
)
=
arcsin
(
x
1
−
y
2
+
y
1
−
x
2
)
{\displaystyle \arcsin(x)+\arcsin(y)=\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}
arccos
(
x
)
+
arccos
(
y
)
=
arccos
(
x
y
−
(
1
−
x
2
)
(
1
−
y
2
)
)
{\displaystyle \arccos(x)+\arccos(y)=\arccos \left(xy-{\sqrt {(1-x^{2})(1-y^{2})}}\right)}
arctan
(
x
)
+
arctan
(
y
)
=
arctan
(
x
+
y
1
−
x
y
)
(
mod
π
)
{\displaystyle \arctan(x)+\arctan(y)=\arctan \left({\frac {x+y}{1-xy}}\right){\pmod {\pi }}}
畢氏定理
編輯
sin
2
(
x
)
+
cos
2
(
x
)
=
1
{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}
1
+
tan
2
(
x
)
=
sec
2
(
x
)
{\displaystyle 1+\tan ^{2}(x)=\sec ^{2}(x)}
1
+
cot
2
(
x
)
=
csc
2
(
x
)
{\displaystyle 1+\cot ^{2}(x)=\csc ^{2}(x)}
倍角公式
編輯
sin
(
2
x
)
=
2
sin
(
x
)
cos
(
x
)
{\displaystyle \sin(2x)=2\sin(x)\cos(x)}
cos
(
2
x
)
=
cos
2
(
x
)
−
sin
2
(
x
)
{\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)}
tan
(
2
x
)
=
2
tan
(
x
)
1
−
tan
2
(
x
)
{\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}}
cos
2
(
x
)
=
1
+
cos
(
2
x
)
2
{\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}
sin
2
(
x
)
=
1
−
cos
(
2
x
)
2
{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}
和角公式
編輯
sin
(
x
+
y
)
=
sin
(
x
)
cos
(
y
)
+
cos
(
x
)
sin
(
y
)
{\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)}
sin
(
x
−
y
)
=
sin
(
x
)
cos
(
y
)
−
cos
(
x
)
sin
(
y
)
{\displaystyle \sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)}
cos
(
x
+
y
)
=
cos
(
x
)
cos
(
y
)
−
sin
(
x
)
sin
(
y
)
{\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)}
cos
(
x
−
y
)
=
cos
(
x
)
cos
(
y
)
+
sin
(
x
)
sin
(
y
)
{\displaystyle \cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)}
sin
(
x
)
+
sin
(
y
)
=
2
sin
(
x
+
y
2
)
cos
(
x
−
y
2
)
{\displaystyle \sin(x)+\sin(y)=2\sin \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)}
sin
(
x
)
−
sin
(
y
)
=
2
cos
(
x
+
y
2
)
sin
(
x
−
y
2
)
{\displaystyle \sin(x)-\sin(y)=2\cos \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)}
cos
(
x
)
+
cos
(
y
)
=
2
cos
(
x
+
y
2
)
cos
(
x
−
y
2
)
{\displaystyle \cos(x)+\cos(y)=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)}
cos
(
x
)
−
cos
(
y
)
=
−
2
sin
(
x
+
y
2
)
sin
(
x
−
y
2
)
{\displaystyle \cos(x)-\cos(y)=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)}
tan
(
x
)
+
tan
(
y
)
=
sin
(
x
+
y
)
cos
(
x
)
cos
(
y
)
{\displaystyle \tan(x)+\tan(y)={\frac {\sin(x+y)}{\cos(x)\cos(y)}}}
tan
(
x
)
−
tan
(
y
)
=
sin
(
x
−
y
)
cos
(
x
)
cos
(
y
)
{\displaystyle \tan(x)-\tan(y)={\frac {\sin(x-y)}{\cos(x)\cos(y)}}}
cot
(
x
)
+
cot
(
y
)
=
sin
(
x
+
y
)
sin
(
x
)
sin
(
y
)
{\displaystyle \cot(x)+\cot(y)={\frac {\sin(x+y)}{\sin(x)\sin(y)}}}
cot
(
x
)
−
cot
(
y
)
=
−
sin
(
x
−
y
)
sin
(
x
)
sin
(
y
)
{\displaystyle \cot(x)-\cot(y)={\frac {-\sin(x-y)}{\sin(x)\sin(y)}}}
積化和差
編輯
cos
(
x
)
cos
(
y
)
=
cos
(
x
+
y
)
+
cos
(
x
−
y
)
2
{\displaystyle \cos(x)\cos(y)={\frac {\cos(x+y)+\cos(x-y)}{2}}}
sin
(
x
)
sin
(
y
)
=
cos
(
x
−
y
)
−
cos
(
x
+
y
)
2
{\displaystyle \sin(x)\sin(y)={\frac {\cos(x-y)-\cos(x+y)}{2}}}
sin
(
x
)
cos
(
y
)
=
sin
(
x
+
y
)
+
sin
(
x
−
y
)
2
{\displaystyle \sin(x)\cos(y)={\frac {\sin(x+y)+\sin(x-y)}{2}}}
cos
(
x
)
sin
(
y
)
=
sin
(
x
+
y
)
−
sin
(
x
−
y
)
2
{\displaystyle \cos(x)\sin(y)={\frac {\sin(x+y)-\sin(x-y)}{2}}}
復指數形式
編輯
e
i
θ
=
c
i
s
θ
=
cos
θ
+
i
sin
θ
{\displaystyle e^{i\theta }=\mathrm {cis} \theta =\cos \theta +i\sin \theta }
sin
θ
=
I
m
(
e
i
θ
)
=
e
i
θ
−
e
−
i
θ
2
i
{\displaystyle \sin \theta =\mathrm {Im} (e^{i\theta })={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}
cos
θ
=
R
e
(
e
i
θ
)
=
e
i
θ
+
e
−
i
θ
2
{\displaystyle \cos \theta =\mathrm {Re} (e^{i\theta })={\frac {e^{i\theta }+e^{-i\theta }}{2}}}
tan
θ
=
sin
θ
cos
θ
=
e
2
i
θ
−
1
i
(
e
2
i
θ
+
1
)
{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {e^{2i\theta }-1}{i(e^{2i\theta }+1)}}}
csc
θ
=
1
sin
θ
=
2
i
e
i
θ
−
e
−
i
θ
{\displaystyle \csc \theta ={\frac {1}{\sin \theta }}={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}
sec
θ
=
1
cos
θ
=
2
e
i
θ
+
e
−
i
θ
{\displaystyle \sec \theta ={\frac {1}{\cos \theta }}={\frac {2}{e^{i\theta }+e^{-i\theta }}}}
cot
θ
=
1
tan
θ
=
i
(
e
2
i
θ
+
1
)
e
2
i
θ
−
1
{\displaystyle \cot \theta ={\frac {1}{\tan \theta }}={\frac {i(e^{2i\theta }+1)}{e^{2i\theta }-1}}}
雙曲函數
編輯
e
x
=
sinh
x
+
cosh
x
{\displaystyle e^{x}=\sinh x+\cosh x}
cosh
2
x
−
sinh
2
x
=
1
{\displaystyle \cosh ^{2}x-\sinh ^{2}x=1}
s
e
c
h
2
x
=
1
−
tanh
2
x
{\displaystyle \mathrm {sech} ^{2}x=1-\tanh ^{2}x}
c
s
c
h
2
x
=
c
o
t
h
2
x
−
1
{\displaystyle \mathrm {csch} ^{2}x=\mathrm {coth} ^{2}x-1}
sinh
x
=
−
i
sin
i
x
=
e
x
−
e
−
x
2
{\displaystyle \sinh x=-i\sin ix={\frac {e^{x}-e^{-x}}{2}}}
cosh
x
=
cos
i
x
=
e
x
+
e
−
x
2
{\displaystyle \cosh x=\cos ix={\frac {e^{x}+e^{-x}}{2}}}
tanh
x
=
−
i
tan
i
x
=
e
x
−
e
−
x
e
x
+
e
−
x
{\displaystyle \tanh x=-i\tan ix={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}
c
s
c
h
x
=
i
csc
i
x
=
2
e
x
−
e
−
x
{\displaystyle \mathrm {csch} x=i\csc ix={\frac {2}{e^{x}-e^{-x}}}}
s
e
c
h
x
=
sec
i
x
=
2
e
x
+
e
−
x
{\displaystyle \mathrm {sech} x=\sec ix={\frac {2}{e^{x}+e^{-x}}}}
c
o
t
h
x
=
i
cot
i
x
=
e
x
+
e
−
x
e
x
−
e
−
x
{\displaystyle \mathrm {coth} x=i\cot ix={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}}
反雙曲函數
編輯
a
r
s
i
n
h
x
=
∫
0
x
1
t
2
+
1
d
t
=
log
(
x
+
x
2
+
1
)
{\displaystyle \mathrm {arsinh} x=\int _{0}^{x}{\frac {1}{\sqrt {t^{2}+1}}}\mathrm {d} t=\log \left(x+{\sqrt {x^{2}+1}}\right)}
a
r
c
o
s
h
x
=
∫
1
x
1
t
2
−
1
d
t
=
log
(
x
+
x
2
−
1
)
{\displaystyle \mathrm {arcosh} x=\int _{1}^{x}{\frac {1}{\sqrt {t^{2}-1}}}\mathrm {d} t=\log \left(x+{\sqrt {x^{2}-1}}\right)}
a
r
t
a
n
h
x
=
∫
0
x
1
1
−
t
2
d
t
=
1
2
log
(
1
+
x
1
−
x
)
{\displaystyle \mathrm {artanh} x=\int _{0}^{x}{\frac {1}{1-t^{2}}}\mathrm {d} t={\frac {1}{2}}\log \left({\frac {1+x}{1-x}}\right)}
a
r
c
c
s
h
x
=
log
(
1
+
1
+
x
2
x
)
{\displaystyle \mathrm {arccsh} x=\log \left({\frac {1+{\sqrt {1+x^{2}}}}{x}}\right)}
a
r
s
e
c
h
x
=
log
(
1
+
1
−
x
2
x
)
{\displaystyle \mathrm {arsech} x=\log \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)}
a
r
c
o
t
h
x
=
1
2
log
(
x
+
1
x
−
1
)
{\displaystyle \mathrm {arcoth} x={\frac {1}{2}}\log \left({\frac {x+1}{x-1}}\right)}
← 附錄
微積分學
求和符號 →
三角函數表
章節導航
:
目錄
·
預備知識
·
極限
·
導數
·
積分
·
極坐標方程與參數方程
·
數列和級數
·
多元函數微積分
·
擴展知識
·
附錄
