These should be at Proofs of trigonometric identities but I dont have the time right now to put them in properly, a copy and paste to there would only increase that articles problems.
These proofs apply directly only to acute angles, but the identities are still correct even when generalized to all angles. In this way, most of the trigonometric identities are deducible from elementary geometry, though many definitions and concepts have to be expanded.
x + y ) = sin(x ) cos(y ) + cos(x ) sin(y )
edit
In the figure the angle x is part of right-angled triangle ABC , and the angle y part of right-angled triangle ACD . Then construct DG perpendicular to AB and construct CE parallel to AB .
angle x = angle BAC = angle ACE = angle CDE .
EG = BC .
sin
(
x
+
y
)
{\displaystyle \sin(x+y)\,}
=
D
G
A
D
{\displaystyle ={\frac {DG}{AD}}\,}
=
E
G
+
D
E
A
D
{\displaystyle ={\frac {EG+DE}{AD}}\,}
=
B
C
+
D
E
A
D
{\displaystyle ={\frac {BC+DE}{AD}}\,}
=
B
C
A
D
+
D
E
A
D
{\displaystyle ={\frac {BC}{AD}}+{\frac {DE}{AD}}\,}
=
B
C
A
D
⋅
A
C
A
C
+
D
E
A
D
⋅
C
D
C
D
{\displaystyle ={\frac {BC}{AD}}\cdot {\frac {AC}{AC}}+{\frac {DE}{AD}}\cdot {\frac {CD}{CD}}\,}
=
B
C
A
C
⋅
A
C
A
D
+
D
E
C
D
⋅
C
D
A
D
{\displaystyle ={\frac {BC}{AC}}\cdot {\frac {AC}{AD}}+{\frac {DE}{CD}}\cdot {\frac {CD}{AD}}\,}
=
sin
(
x
)
cos
(
y
)
+
cos
(
x
)
sin
(
y
)
.
{\displaystyle =\sin(x)\cos(y)+\cos(x)\sin(y).\,}
x + y ) = cos(x ) cos(y ) − sin(x ) sin(y )
edit
Using the above figure:
cos
(
x
+
y
)
{\displaystyle \cos(x+y)\,}
=
A
G
A
D
{\displaystyle ={\frac {AG}{AD}}\,}
=
A
B
−
G
B
A
D
{\displaystyle ={\frac {AB-GB}{AD}}\,}
=
A
B
−
E
C
A
D
{\displaystyle ={\frac {AB-EC}{AD}}\,}
=
A
B
A
D
−
E
C
A
D
{\displaystyle ={\frac {AB}{AD}}-{\frac {EC}{AD}}\,}
=
A
B
A
D
⋅
A
C
A
C
−
E
C
A
D
⋅
C
D
C
D
{\displaystyle ={\frac {AB}{AD}}\cdot {\frac {AC}{AC}}-{\frac {EC}{AD}}\cdot {\frac {CD}{CD}}\,}
=
A
B
A
C
⋅
A
C
A
D
−
E
C
C
D
⋅
C
D
A
D
{\displaystyle ={\frac {AB}{AC}}\cdot {\frac {AC}{AD}}-{\frac {EC}{CD}}\cdot {\frac {CD}{AD}}\,}
=
cos
(
x
)
cos
(
y
)
−
sin
(
x
)
sin
(
y
)
.
{\displaystyle =\cos(x)\cos(y)-\sin(x)\sin(y).\,}
The formulæ for cos(x − y ) and sin(x − y ) are easily proven using the formulæ for cos(x + y ) and sin(x + y ), respectively
x − y ) = sin(x ) cos(y ) − cos(x ) sin(y )
edit
To begin, we substitute y with −y into the sin(x + y ) formula:
sin
(
x
+
(
−
y
)
)
=
sin
(
x
)
cos
(
−
y
)
+
cos
(
x
)
sin
(
−
y
)
.
{\displaystyle \!\sin(x+(-y))=\sin(x)\cos(-y)+\cos(x)\sin(-y).}
Using the fact that sine is an odd function and cosine is an even function , we get
sin
(
x
−
y
)
=
sin
(
x
)
cos
(
y
)
−
cos
(
x
)
sin
(
y
)
.
{\displaystyle \!\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y).}
x − y ) = cos(x ) cos(y ) + sin(x ) sin(y )
edit
To begin, we substitute y with −y into the cos(x + y ) formula:
cos
(
x
+
(
−
y
)
)
=
cos
(
x
)
cos
(
−
y
)
−
sin
(
x
)
sin
(
−
y
)
.
{\displaystyle \!\cos(x+(-y))=\cos(x)\cos(-y)-\sin(x)\sin(-y).}
Using the fact that sine is an odd function and cosine is an even function, we get
cos
(
x
−
y
)
=
cos
(
x
)
cos
(
y
)
+
sin
(
x
)
sin
(
y
)
.
{\displaystyle \!\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y).}
edit
I removed the following from the article because they are not identities of trigonometric functions. I intend to create a parallel article List of hyperbolic identities which can contain the.
sinh
(
θ
)
=
−
i
sin
(
i
θ
)
=
e
θ
−
e
−
θ
2
{\displaystyle \operatorname {sinh} (\theta )=-i\sin(i\theta )={\frac {e^{\theta }-e^{-\theta }}{2}}\,}
cosh
(
θ
)
=
cos
(
i
θ
)
=
e
θ
+
e
−
θ
2
{\displaystyle \operatorname {cosh} (\theta )=\cos(i\theta )={\frac {e^{\theta }+e^{-\theta }}{2}}\,}
tanh
(
θ
)
=
sinh
(
θ
)
cosh
(
θ
)
=
e
θ
−
e
−
θ
e
θ
+
e
−
θ
=
e
2
θ
−
1
e
2
θ
+
1
{\displaystyle \operatorname {tanh} (\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}}\,}
coth
(
θ
)
=
cosh
(
θ
)
sinh
(
θ
)
=
e
θ
+
e
−
θ
e
θ
−
e
−
θ
=
e
2
θ
+
1
e
2
θ
−
1
{\displaystyle \operatorname {coth} (\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}}\,}
sech
(
θ
)
=
1
cosh
(
θ
)
=
sec
(
i
θ
)
=
2
e
θ
+
e
−
θ
{\displaystyle \operatorname {sech} (\theta )={\frac {1}{\operatorname {cosh} (\theta )}}=\operatorname {sec} (i\theta )={\frac {2}{e^{\theta }+e^{-\theta }}}\,}
csch
(
θ
)
=
1
sinh
(
θ
)
=
i
cos
(
i
θ
)
=
2
e
θ
−
e
−
θ
{\displaystyle \operatorname {csch} (\theta )={\frac {1}{\operatorname {sinh} (\theta )}}=i\cos(i\theta )={\frac {2}{e^{\theta }-e^{-\theta }}}\,}
versinh
(
θ
)
=
1
−
cosh
(
θ
)
=
1
−
cos
(
i
θ
)
=
1
−
e
θ
+
e
−
θ
2
{\displaystyle \operatorname {versinh} (\theta )=1-\operatorname {cosh} (\theta )=1-\cos(i\theta )=1-{\frac {e^{\theta }+e^{-\theta }}{2}}\,}
vercosh
(
θ
)
=
1
−
sinh
(
θ
)
=
1
+
i
sin
(
i
θ
)
=
1
−
e
θ
−
e
−
θ
2
{\displaystyle \operatorname {vercosh} (\theta )=1-\operatorname {sinh} (\theta )=1+i\sin(i\theta )=1-{\frac {e^{\theta }-e^{-\theta }}{2}}\,}
exsech
(
θ
)
=
sech
(
θ
)
−
1
=
1
cosh
(
θ
)
−
1
=
2
e
θ
+
e
−
θ
−
1
{\displaystyle \operatorname {exsech} (\theta )=\operatorname {sech} (\theta )-1={\frac {1}{\operatorname {cosh} (\theta )}}-1={\frac {2}{e^{\theta }+e^{-\theta }}}-1\,}
excsch
(
θ
)
=
csch
(
θ
)
−
1
=
1
sinh
(
θ
)
−
1
=
2
e
θ
−
e
−
θ
−
1
{\displaystyle \operatorname {excsch} (\theta )=\operatorname {csch} (\theta )-1={\frac {1}{\operatorname {sinh} (\theta )}}-1={\frac {2}{e^{\theta }-e^{-\theta }}}-1\,}
arcsinh
(
θ
)
=
ln
(
θ
+
θ
2
+
1
)
{\displaystyle \operatorname {arcsinh} (\theta )=\ln(\theta +{\sqrt {\theta ^{2}+1}})\,}
arccosh
(
θ
)
=
ln
(
θ
+
θ
2
−
1
)
{\displaystyle \operatorname {arccosh} (\theta )=\ln(\theta +{\sqrt {\theta ^{2}-1}})\,}
arctanh
(
θ
)
=
ln
(
i
+
θ
i
−
θ
)
2
{\displaystyle \operatorname {arctanh} (\theta )={\frac {\ln({\frac {i+\theta }{i-\theta }})}{2}}\,}
arccoth
(
θ
)
=
arctanh
(
−
θ
)
=
ln
(
i
−
θ
i
+
θ
)
2
{\displaystyle \operatorname {arccoth} (\theta )=\operatorname {arctanh} (-\theta )={\frac {\ln({\frac {i-\theta }{i+\theta }})}{2}}\,}
arcsech
(
θ
)
=
arccosh
(
θ
−
1
)
=
ln
(
θ
−
1
+
θ
−
2
−
1
)
{\displaystyle \operatorname {arcsech} (\theta )=\operatorname {arccosh} (\theta ^{-1})=\ln(\theta ^{-1}+{\sqrt {\theta ^{-2}-1}})\,}
arccsch
(
θ
)
=
arcsinh
(
θ
−
1
)
=
ln
(
θ
−
1
+
θ
−
2
+
1
)
{\displaystyle \operatorname {arccsch} (\theta )=\operatorname {arcsinh} (\theta ^{-1})=\ln(\theta ^{-1}+{\sqrt {\theta ^{-2}+1}})\,}
arcversinh
(
θ
)
=
arccos
(
1
−
θ
)
i
{\displaystyle \operatorname {arcversinh} (\theta )={\frac {\operatorname {arccos} (1-\theta )}{i}}\,}
arcvercosh
(
θ
)
=
arcsin
θ
−
1
i
i
{\displaystyle \operatorname {arcvercosh} (\theta )={\frac {\operatorname {arcsin} {\frac {\theta -1}{i}}}{i}}\,}
arcexsech
(
θ
)
=
arcsech
(
θ
+
1
)
=
arccosh
(
(
θ
+
1
)
−
1
)
=
ln
(
(
θ
+
1
)
−
1
+
(
θ
+
1
)
−
2
−
1
)
{\displaystyle \operatorname {arcexsech} (\theta )=\operatorname {arcsech} (\theta +1)=\operatorname {arccosh} ((\theta +1)^{-1})=\ln((\theta +1)^{-1}+{\sqrt {(\theta +1)^{-2}-1}})\,}
arcexcsch
(
θ
)
=
arccsch
(
θ
+
1
)
=
arcsinh
(
(
θ
+
1
)
−
1
)
=
ln
(
(
θ
+
1
)
−
1
+
(
θ
+
1
)
−
2
+
1
)
{\displaystyle \operatorname {arcexcsch} (\theta )=\operatorname {arccsch} (\theta +1)=\operatorname {arcsinh} ((\theta +1)^{-1})=\ln((\theta +1)^{-1}+{\sqrt {(\theta +1)^{-2}+1}})\,}
cish
(
θ
)
=
cosh
(
θ
)
+
i
sinh
(
θ
)
=
e
θ
+
e
−
θ
2
+
i
e
θ
−
e
−
θ
2
=
cos
(
i
θ
)
+
sin
(
i
θ
)
{\displaystyle \operatorname {cish} (\theta )=\operatorname {cosh} (\theta )+i\ \operatorname {sinh} (\theta )={\frac {e^{\theta }+e^{-\theta }}{2}}+i{\frac {e^{\theta }-e^{-\theta }}{2}}=\operatorname {cos} (i\theta )+\operatorname {sin} (i\theta )\,}
arccish
(
θ
)
=
arcsin
(
θ
2
−
1
)
2
i
=
−
ln
(
i
(
θ
2
−
1
)
+
1
−
(
θ
2
−
1
)
2
)
2
{\displaystyle \operatorname {arccish} (\theta )={\frac {\operatorname {arcsin} (\theta ^{2}-1)}{2i}}={\frac {-\ln(i(\theta ^{2}-1)+{\sqrt {1-(\theta ^{2}-1)^{2}}})}{2}}\,}
edit
I removed the following from the article because I dont see that this is appropriate content for an enclyclopedia, it might be found in a maths textbook, in which case the following cold be used in a wikibook.
tan
(
θ
)
{\displaystyle \operatorname {tan} (\theta )\,}
=
sin
(
θ
)
cos
(
θ
)
{\displaystyle ={\frac {\operatorname {sin} (\theta )}{\operatorname {cos} (\theta )}}\,}
=
e
i
θ
−
e
−
i
θ
2
i
e
i
θ
+
e
−
i
θ
2
{\displaystyle ={\frac {\frac {e^{i\theta }-e^{-i\theta }}{2i}}{\frac {e^{i\theta }+e^{-i\theta }}{2}}}\,}
=
2
(
e
i
θ
−
e
−
i
θ
)
2
i
(
e
i
θ
+
e
−
i
θ
)
{\displaystyle ={\frac {2(e^{i\theta }-e^{-i\theta })}{2i(e^{i\theta }+e^{-i\theta })}}\,}
=
e
i
θ
−
e
−
i
θ
i
(
e
i
θ
+
e
−
i
θ
)
{\displaystyle ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}
cot
(
θ
)
=
cos
(
θ
)
sin
(
θ
)
=
1
tan
θ
=
(
e
i
θ
+
e
−
i
θ
2
)
(
e
i
θ
−
e
−
i
θ
2
i
)
=
2
i
(
e
i
θ
+
e
−
i
θ
)
2
(
e
i
θ
−
e
−
i
θ
)
=
i
(
e
i
θ
+
e
−
i
θ
)
e
i
θ
−
e
−
i
θ
{\displaystyle \cot(\theta )={\frac {\cos(\theta )}{\sin(\theta )}}={\frac {1}{\tan \theta }}={\frac {\left({\frac {e^{i\theta }+e^{-i\theta }}{2}}\right)}{\left({\frac {e^{i\theta }-e^{-i\theta }}{2i}}\right)}}={\frac {2i(e^{i\theta }+e^{-i\theta })}{2(e^{i\theta }-e^{-i\theta })}}={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}
sec
(
θ
)
=
1
cos
(
θ
)
=
1
(
e
i
θ
+
e
−
i
θ
2
)
=
2
e
i
θ
+
e
−
i
θ
{\displaystyle {\begin{aligned}\operatorname {sec} (\theta )&{}={\frac {1}{\operatorname {cos} (\theta )}}\\\\&{}={\frac {1}{({\frac {e^{i\theta }+e^{-i\theta }}{2}})}}\\\\&{}={\frac {2}{e^{i\theta }+e^{-i\theta }}}\end{aligned}}}
csc
(
θ
)
=
1
sin
(
θ
)
=
1
(
e
i
θ
−
e
−
i
θ
2
i
)
=
2
i
e
i
θ
−
e
−
i
θ
{\displaystyle {\begin{aligned}\operatorname {csc} (\theta )&{}={\frac {1}{\operatorname {sin} (\theta )}}\\\\&{}={\frac {1}{({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}}\\\\&{}={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\end{aligned}}}
cis
(
x
)
{\displaystyle \operatorname {cis} (x)\,}
=
cos
(
x
)
+
i
sin
(
x
)
{\displaystyle =\operatorname {cos} (x)+i\ \operatorname {sin} (x)\,}
=
e
i
x
+
e
−
i
x
2
+
i
e
i
x
−
e
−
i
x
2
i
{\displaystyle ={\frac {e^{ix}+e^{-ix}}{2}}+i{\frac {e^{ix}-e^{-ix}}{2i}}\,}
=
e
i
x
+
e
−
i
x
2
+
e
i
x
−
e
−
i
x
2
{\displaystyle ={\frac {e^{ix}+e^{-ix}}{2}}+{\frac {e^{ix}-e^{-ix}}{2}}\,}
=
(
e
i
x
+
e
−
i
x
)
+
(
e
i
x
−
e
−
i
x
)
2
{\displaystyle ={\frac {(e^{ix}+e^{-ix})+(e^{ix}-e^{-ix})}{2}}\,}
=
e
i
x
+
e
−
i
x
+
e
i
x
−
e
−
i
x
2
{\displaystyle ={\frac {e^{ix}+e^{-ix}+e^{ix}-e^{-ix}}{2}}\,}
=
e
i
x
+
e
i
x
+
e
−
i
x
−
e
−
i
x
2
{\displaystyle ={\frac {e^{ix}+e^{ix}+e^{-ix}-e^{-ix}}{2}}\,}
=
e
i
x
+
e
i
x
+
0
2
{\displaystyle ={\frac {e^{ix}+e^{ix}+0}{2}}\,}
=
e
i
x
+
e
i
x
2
{\displaystyle ={\frac {e^{ix}+e^{ix}}{2}}\,}
=
2
(
e
i
x
)
2
{\displaystyle ={\frac {2(e^{ix})}{2}}}
=
e
i
x
{\displaystyle =e^{ix}\,}
tanh
(
x
)
=
sinh
(
x
)
cosh
(
x
)
=
(
e
x
−
e
−
x
2
)
(
e
x
+
e
−
x
2
)
=
(
e
x
−
e
−
x
2
)
e
x
+
e
−
x
2
=
e
x
−
e
−
x
e
x
+
e
−
x
=
(
e
x
−
e
−
x
e
−
x
)
(
e
x
+
e
−
x
e
−
x
)
=
e
x
e
−
x
−
1
e
x
e
−
x
+
1
=
e
x
e
x
−
1
e
x
e
x
+
1
=
(
e
x
)
2
−
1
(
e
x
)
2
+
1
=
e
2
x
−
1
e
2
x
+
1
.
{\displaystyle {\begin{aligned}\operatorname {tanh} (x)&{}={\frac {\operatorname {sinh} (x)}{\operatorname {cosh} (x)}}\\\\&{}={\dfrac {\left({\dfrac {e^{x}-e^{-x}}{2}}\right)}{\left({\dfrac {e^{x}+e^{-x}}{2}}\right)}}\\\\&{}={\dfrac {\left({\dfrac {e^{x}-e^{-x}}{2}}\right)}{e^{x}+e^{-x}}}2\\\\&{}={\dfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}\\\\&{}={\dfrac {\left({\dfrac {e^{x}-e^{-x}}{e^{-x}}}\right)}{\left({\dfrac {e^{x}+e^{-x}}{e^{-x}}}\right)}}\\\\&{}={\dfrac {{\dfrac {e^{x}}{e^{-x}}}-1}{{\dfrac {e^{x}}{e^{-x}}}+1}}\\\\&{}={\dfrac {e^{x}e^{x}-1}{e^{x}e^{x}+1}}\\\\&{}={\dfrac {(e^{x})^{2}-1}{(e^{x})^{2}+1}}\\\\&{}={\frac {e^{2x}-1}{e^{2x}+1}}.\end{aligned}}}
coth
(
x
)
=
cosh
(
x
)
sinh
(
x
)
=
(
e
x
+
e
−
x
2
)
(
e
x
−
e
−
x
2
)
=
e
x
+
e
−
x
e
x
−
e
−
x
=
(
e
x
+
e
−
x
e
−
x
)
(
e
x
−
e
−
x
e
−
x
)
=
e
x
e
−
x
+
1
e
x
e
−
x
−
1
=
e
x
e
x
+
1
e
x
e
x
−
1
=
(
e
x
)
2
+
1
(
e
x
)
2
−
1
=
e
2
x
+
1
e
2
x
−
1
{\displaystyle {\begin{aligned}\coth(x)&{}={\frac {\cosh(x)}{\sinh(x)}}\\\\&{}={\frac {({\frac {e^{x}+e^{-x}}{2}})}{({\frac {e^{x}-e^{-x}}{2}})}}\\\\&{}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}\\\\&{}={\frac {({\frac {e^{x}+e^{-x}}{e^{-x}}})}{({\frac {e^{x}-e^{-x}}{e^{-x}}})}}\\\\&{}={\frac {{\frac {e^{x}}{e^{-x}}}+1}{{\frac {e^{x}}{e^{-x}}}-1}}\\\\&{}={\frac {e^{x}e^{x}+1}{e^{x}e^{x}-1}}\\\\&{}={\frac {(e^{x})^{2}+1}{(e^{x})^{2}-1}}\\\\&{}={\frac {e^{2x}+1}{e^{2x}-1}}\end{aligned}}}
sech
(
x
)
=
1
cosh
(
x
)
=
1
(
e
x
+
e
−
x
2
)
=
2
e
x
+
e
−
x
{\displaystyle {\begin{aligned}\operatorname {sech} (x)&{}={\frac {1}{\cosh(x)}}\\\\&{}={\frac {1}{\left({\dfrac {e^{x}+e^{-x}}{2}}\right)}}\\\\&{}={\frac {2}{e^{x}+e^{-x}}}\end{aligned}}}
edit
sech
(
x
)
=
1
cosh
(
x
)
=
1
cos
(
i
x
)
=
1
(
e
i
2
x
+
e
−
i
2
x
2
)
=
2
e
i
2
x
+
e
−
i
2
x
=
2
e
−
x
+
e
−
(
−
1
)
x
=
2
e
−
x
+
e
x
=
2
e
x
+
e
−
x
{\displaystyle {\begin{aligned}\operatorname {sech} (x)&{}={\frac {1}{\operatorname {cosh} (x)}}\\&{}={\frac {1}{\operatorname {cos} (ix)}}\\&{}={\frac {1}{({\frac {e^{i^{2}x}+e^{-i^{2}x}}{2}})}}\\&{}={\frac {2}{e^{i^{2}x}+e^{-i^{2}x}}}\\&{}={\frac {2}{e^{-x}+e^{-(-1)x}}}\\&{}={\frac {2}{e^{-x}+e^{x}}}\\&{}={\frac {2}{e^{x}+e^{-x}}}\end{aligned}}}
csch
(
x
)
=
1
sinh
(
x
)
=
1
(
e
x
−
e
−
x
2
)
=
1
e
x
−
e
−
x
2
=
2
e
x
−
e
−
x
{\displaystyle {\begin{aligned}\operatorname {csch} (x)&{}={\frac {1}{\operatorname {sinh} (x)}}\\&{}={\frac {1}{({\frac {e^{x}-e^{-x}}{2}})}}\\&{}={\frac {1}{e^{x}-e^{-x}}}2\\&{}={\frac {2}{e^{x}-e^{-x}}}\end{aligned}}}
edit
csch
(
x
)
=
{\displaystyle \operatorname {csch} (x)=\,}
=
1
sinh
(
x
)
=
{\displaystyle ={\frac {1}{\operatorname {sinh} (x)}}=\,}
=
1
−
i
sin
(
i
x
)
=
{\displaystyle ={\frac {1}{-i\ \operatorname {sin} (ix)}}=\,}
=
1
−
i
e
i
2
x
−
e
−
i
2
x
2
i
=
{\displaystyle ={\frac {1}{-i{\frac {e^{i^{2}x}-e^{-i^{2}x}}{2i}}}}=\,}
=
1
−
i
e
−
x
−
e
x
2
i
=
{\displaystyle ={\frac {1}{-i{\frac {e^{-x}-e^{x}}{2i}}}}=\,}
=
1
−
i
−
e
x
+
e
−
x
2
i
=
{\displaystyle ={\frac {1}{-i{\frac {-e^{x}+e^{-x}}{2i}}}}=\,}
=
1
i
e
x
−
e
−
x
2
i
=
{\displaystyle ={\frac {1}{i{\frac {e^{x}-e^{-x}}{2i}}}}=\,}
=
1
e
x
−
e
−
x
2
=
{\displaystyle ={\frac {1}{\frac {e^{x}-e^{-x}}{2}}}=\,}
=
1
e
x
−
e
−
x
2
=
{\displaystyle ={\frac {1}{e^{x}-e^{-x}}}2=\,}
=
2
e
x
−
e
−
x
{\displaystyle ={\frac {2}{e^{x}-e^{-x}}}\,}
sec
(
x
)
=
θ
{\displaystyle \operatorname {sec} (x)=\theta \,}
1
cos
(
x
)
=
θ
{\displaystyle {\frac {1}{\operatorname {cos} (x)}}=\theta \,}
cos
(
x
)
=
θ
−
1
{\displaystyle \operatorname {cos} (x)=\theta ^{-1}\,}
x
=
arccos
(
θ
−
1
)
{\displaystyle x=\operatorname {arccos} (\theta ^{-1})\,}
csc
(
x
)
=
θ
{\displaystyle \operatorname {csc} (x)=\theta \,}
1
sin
(
x
)
=
θ
{\displaystyle {\frac {1}{\operatorname {sin} (x)}}=\theta \,}
sin
(
x
)
=
θ
−
1
{\displaystyle \operatorname {sin} (x)=\theta ^{-1}\,}
x
=
arcsin
(
θ
−
1
)
{\displaystyle x=\operatorname {arcsin} (\theta ^{-1})\,}
edit
versin
(
x
)
=
θ
{\displaystyle \operatorname {versin} (x)=\theta \,}
1
−
cos
(
x
)
=
θ
{\displaystyle 1-\operatorname {cos} (x)=\theta \,}
cos
(
x
)
=
1
−
θ
{\displaystyle \operatorname {cos} (x)=1-\theta \,}
x
=
arccos
(
1
−
θ
)
{\displaystyle x=\operatorname {arccos} (1-\theta )\,}
exsec
(
x
)
=
θ
{\displaystyle \operatorname {exsec} (x)=\theta \,}
sec
(
x
)
−
1
=
θ
{\displaystyle \operatorname {sec} (x)-1=\theta \,}
sec
(
x
)
=
θ
+
1
{\displaystyle \operatorname {sec} (x)=\theta +1\,}
x
=
arcsec
(
θ
+
1
)
{\displaystyle x=\operatorname {arcsec} (\theta +1)\,}
excsc
(
x
)
=
θ
{\displaystyle \operatorname {excsc} (x)=\theta \,}
csc
(
x
)
−
1
=
θ
{\displaystyle \operatorname {csc} (x)-1=\theta \,}
csc
(
x
)
=
θ
+
1
{\displaystyle \operatorname {csc} (x)=\theta +1\,}
x
=
arccsc
(
θ
+
1
)
{\displaystyle x=\operatorname {arccsc} (\theta +1)\,}
θ
=
sech
(
x
)
{\displaystyle \theta =\operatorname {sech} (x)\,}
θ
=
1
cosh
(
x
)
{\displaystyle \theta ={\frac {1}{\operatorname {cosh} (x)}}\,}
1
θ
=
cosh
(
x
)
{\displaystyle {\frac {1}{\theta }}=\operatorname {cosh} (x)\,}
θ
−
1
=
cosh
(
x
)
{\displaystyle \theta ^{-1}=\operatorname {cosh} (x)\,}
arccosh
(
θ
−
1
)
=
x
{\displaystyle \operatorname {arccosh} (\theta ^{-1})=x\,}
θ
=
csch
(
x
)
{\displaystyle \theta =\operatorname {csch} (x)\,}
θ
=
1
sinh
(
x
)
{\displaystyle \theta ={\frac {1}{\operatorname {sinh} (x)}}\,}
1
θ
=
sinh
(
x
)
{\displaystyle {\frac {1}{\theta }}=\operatorname {sinh} (x)\,}
θ
−
1
=
sinh
(
x
)
{\displaystyle \theta ^{-1}=\operatorname {sinh} (x)\,}
arcsinh
(
θ
−
1
)
=
x
{\displaystyle \operatorname {arcsinh} (\theta ^{-1})=x\,}
edit
θ
=
versinh
(
x
)
{\displaystyle \theta =\operatorname {versinh} (x)\,}
θ
=
1
−
cosh
(
x
)
{\displaystyle \theta =1-\operatorname {cosh} (x)\,}
θ
=
1
−
cos
(
i
x
)
{\displaystyle \theta =1-\operatorname {cos} (ix)\,}
1
−
θ
=
cos
(
i
x
)
{\displaystyle 1-\theta =\operatorname {cos} (ix)\,}
arccos
(
1
−
θ
)
=
i
x
{\displaystyle \operatorname {arccos} (1-\theta )=ix\,}
x
=
arccos
(
1
−
θ
)
i
{\displaystyle x={\frac {\operatorname {arccos} (1-\theta )}{i}}\,}
edit
θ
=
exsech
(
x
)
{\displaystyle \theta =\operatorname {exsech} (x)\,}
θ
=
sech
(
x
)
−
1
{\displaystyle \theta =\operatorname {sech} (x)-1\,}
θ
+
1
=
sech
(
x
)
{\displaystyle \theta +1=\operatorname {sech} (x)\,}
x
=
arcsech
(
θ
+
1
)
{\displaystyle x=\operatorname {arcsech} (\theta +1)\,}
edit
θ
=
excsch
(
x
)
{\displaystyle \theta =\operatorname {excsch} (x)\,}
θ
=
csch
(
x
)
−
1
{\displaystyle \theta =\operatorname {csch} (x)-1\,}
θ
+
1
=
csch
(
x
)
{\displaystyle \theta +1=\operatorname {csch} (x)\,}
x
=
arccsch
(
θ
+
1
)
{\displaystyle x=\operatorname {arccsch} (\theta +1)\,}
θ
=
cis
(
x
)
{\displaystyle \theta =\operatorname {cis} (x)\,}
θ
=
e
i
x
{\displaystyle \theta =e^{ix}\,}
ln
θ
=
ln
e
i
x
{\displaystyle \ln \theta =\ln e^{ix}\,}
ln
θ
=
i
x
{\displaystyle \ln \theta =ix\,}
ln
θ
i
=
i
x
i
{\displaystyle {\frac {\ln \theta }{i}}={\frac {ix}{i}}\,}
ln
θ
i
=
x
{\displaystyle {\frac {\ln \theta }{i}}=x\,}
θ
=
cish
(
x
)
{\displaystyle \theta =\operatorname {cish} (x)\,}
θ
=
cosh
(
x
)
+
i
sinh
(
x
)
{\displaystyle \theta =\operatorname {cosh} (x)+i\ \operatorname {sinh} (x)\,}
θ
2
=
(
cosh
(
x
)
+
i
sinh
(
x
)
)
2
{\displaystyle \theta ^{2}=(\operatorname {cosh} (x)+i\ \operatorname {sinh} (x))^{2}\,}
θ
2
=
c
o
s
h
2
(
x
)
+
2
i
cosh
(
x
)
sinh
(
x
)
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+2i\ \operatorname {cosh} (x)\operatorname {sinh} (x)-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
+
2
i
cos
(
i
x
)
sinh
(
x
)
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+2i\ \operatorname {cos} (ix)\operatorname {sinh} (x)-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
+
2
cos
(
i
x
)
sin
(
i
x
)
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+2\ \operatorname {cos} (ix)\operatorname {sin} (ix)-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
+
sin
(
i
x
+
i
x
)
−
sin
(
i
x
−
i
x
)
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (ix+ix)-\operatorname {sin} (ix-ix)-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
+
sin
(
2
i
x
)
−
sin
(
0
)
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (2ix)-\operatorname {sin} (0)-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
+
sin
(
2
i
x
)
−
0
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (2ix)-0-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
+
sin
(
2
i
x
)
−
s
i
n
h
2
(
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (2ix)-\operatorname {sinh^{2}} (x)\,}
θ
2
=
c
o
s
h
2
(
x
)
−
s
i
n
h
2
(
x
)
+
sin
(
2
i
x
)
{\displaystyle \theta ^{2}=\operatorname {cosh^{2}} (x)-\operatorname {sinh^{2}} (x)+\operatorname {sin} (2ix)\,}
θ
2
=
1
+
sin
(
2
i
x
)
{\displaystyle \theta ^{2}=1+\operatorname {sin} (2ix)\,}
θ
2
−
1
=
sin
(
2
i
x
)
{\displaystyle \theta ^{2}-1=\operatorname {sin} (2ix)\,}
arcsin
(
θ
2
−
1
)
=
2
i
x
{\displaystyle \operatorname {arcsin} (\theta ^{2}-1)=2ix\,}
x
=
arcsin
(
θ
2
−
1
)
2
i
{\displaystyle x={\frac {\operatorname {arcsin} (\theta ^{2}-1)}{2i}}\,}
