Sign up Log in
Inverse Trigonometric Identities
Quiz
Inverse Trigonometric Identities
Relevant For...
Geometry>
Sum and Difference Trigonometric Formulas
Omkar Kulkarni, Pranjal Jain, Jimin Khim, and Β 1 other Β contributed
Before reading this, make sure you are familiar with inverse trigonometric functions.
The following inverse trigonometric identities give an angle in different ratios. Before the more complicated identities come some seemingly obvious ones. Be observant of the conditions the identities call for.
β
1
(
β
x
)
=
β
β
1
x
,
β£
x
β£
β€
1
β
1
(
β
x
)
=
Ο
β
β
1
x
,
β£
x
β£
β€
1
β
1
(
β
x
)
=
β
β
1
x
,
x
β
R
β
1
(
β
x
)
=
Ο
β
β
1
x
,
x
β
R
β
1
x
=
β
1
(
1
x
)
,
β£
x
β£
β₯
1
β
1
x
=
β
1
(
1
x
)
,
β£
x
β£
β₯
1
β
1
x
=
β
1
(
1
x
)
,
x
>
0
β
1
x
=
Ο
+
β
1
(
1
x
)
,
x
<
0
β
1
x
+
β
1
x
=
Ο
2
,
β£
x
β£
β€
1
β
1
x
+
β
1
x
=
Ο
2
,
β£
x
β£
β₯
1
sin
β1
(βx)
cos
β1
(βx)
tan
β1
(βx)
cot
β1
(βx)
csc
β1
x
sec
β1
x
cot
β1
x
cot
β1
x
sin
β1
x+cos
β1
x
csc
β1
x+sec
β1
x
β
Β
=βsin
β1
x,
=Οβcos
β1
x,
=βtan
β1
x,
=Οβcot
β1
x,
=sin
β1
(
x
1
β
),
=cos
β1
(
x
1
β
),
=tan
β1
(
x
1
β
),
=Ο+tan
β1
(
x
1
β
),
=
2
Ο
β
,
=
2
Ο
β
,
β
Β
β£xβ£β€1
β£xβ£β€1
xβR
xβR
β£xβ£β₯1
β£xβ£β₯1
x>0
x<0
β£xβ£β€1
β£xβ£β₯1
β
Now for the more complicated identities. These come handy very often, and can easily be derived using the basic trigonometric identities.
β
1
x
=
β
1
(
1
β
x
2
)
,
x
β₯
0
β
1
x
=
β
1
(
1
β
x
2
)
,
x
β₯
0
β
1
x
=
Ο
β
β
1
(
1
β
x
2
)
,
x
<
0
β
1
x
+
β
1
y
=
β
1
(
x
+
y
1
β
x
y
)
,
x
y
<
1
β
1
x
+
β
1
y
=
Ο
+
β
1
(
x
+
y
1
β
x
y
)
,
x
y
>
1
β
1
x
β
β
1
y
=
β
1
(
x
β
y
1
+
x
y
)
β
1
x
=
β
1
(
x
1
β
x
2
)
,
x
β
(
0
,
1
)
β
1
x
=
β
1
(
1
β
x
2
x
)
,
x
β
(
0
,
1
)
β
1
x
=
β
1
(
x
x
2
+
1
)
,
x
>
0
β
1
x
=
β
1
(
1
x
2
+
1
)
,
x
>
0
sin
β1
x
cos
β1
x
cos
β1
x
tan
β1
x+tan
β1
y
tan
β1
x+tan
β1
y
tan
β1
xβtan
β1
y
sin
β1
x
cos
β1
x
tan
β1
x
tan
β1
x
β
Β
=cos
β1
(
1βx
2
β
),
=sin
β1
(
1βx
2
β
),
=Οβsin
β1
(
1βx
2
β
),
=tan
β1
(
1βxy
x+y
β
),
=Ο+tan
β1
(
1βxy
x+y
β
),
=tan
β1
(
1+xy
xβy
β
)
=tan
β1
(
1βx
2
β
x
β
),
=tan
β1
(
x
1βx
2
β
β
),
=sin
β1
(
x
2
+1
β
x
β
),
=cos
β1
(
x
2
+1
β
1
β
),
β
Β
xβ₯0
xβ₯0
x<0
xy<1
xy>1
xβ(0,1)
xβ(0,1)
x>0
x>0
β
Find the value of
x
x for which
sin
β‘
(
β
1
(
1
+
x
)
)
=
cos
β‘
(
β
1
(
x
)
)
.
sin(cot
β1
(1+x))=cos(tan
β1
(x)).
We have
β
1
(
1
+
x
)
=
β
1
(
1
1
+
(
1
+
x
)
2
)
β
1
x
=
β
1
(
1
x
2
β
1
)
.
cot
β1
(1+x)
tan
β1
x
β
Β
=sin
β1
(
1+(1+x)
2
β
1
β
)
=cos
β1
(
x
2
β1
β
1
β
).
β
Therefore, we have
sin
β‘
(
β
1
(
1
+
x
)
)
=
cos
β‘
(
β
1
(
x
)
)
sin
β‘
(
β
1
(
1
1
+
(
1
+
x
)
2
)
)
=
cos
β‘
(
β
1
(
1
x
2
β
1
)
)
1
1
+
(
1
+
x
)
2
=
1
x
2
+
1
x
2
+
1
=
(
x
+
1
)
2
+
1
x
2
+
2
x
+
1
=
x
2
x
=
β
1
2
.
β‘
sin(cot
β1
(1+x))
sin(sin
β1
(
1+(1+x)
2
β
1
β
))
1+(1+x)
2
β
1
β
x
2
+1
x
2
+2x+1
x
β
Β
=cos(tan
β1
(x))
=cos(cos
β1
(
x
2
β1
β
1
β
))
=
x
2
+1
β
1
β
=(x+1)
2
+1
=x
2
=β
2
1
β
. Β
β‘
β
β
Cite as: Inverse Trigonometric Identities. Brilliant.org. Retrieved 22:45, February 27, 2020, from https://brilliant.org/wiki/inverse-trigonometric-identities/
Get more Brilliant. Sign upSign up Log in
Inverse Trigonometric Identities
Quiz
Inverse Trigonometric Identities
Relevant For...
Geometry>
Sum and Difference Trigonometric Formulas
Omkar Kulkarni, Pranjal Jain, Jimin Khim, and Β 1 other Β contributed
Before reading this, make sure you are familiar with inverse trigonometric functions.
The following inverse trigonometric identities give an angle in different ratios. Before the more complicated identities come some seemingly obvious ones. Be observant of the conditions the identities call for.
β
1
(
β
x
)
=
β
β
1
x
,
β£
x
β£
β€
1
β
1
(
β
x
)
=
Ο
β
β
1
x
,
β£
x
β£
β€
1
β
1
(
β
x
)
=
β
β
1
x
,
x
β
R
β
1
(
β
x
)
=
Ο
β
β
1
x
,
x
β
R
β
1
x
=
β
1
(
1
x
)
,
β£
x
β£
β₯
1
β
1
x
=
β
1
(
1
x
)
,
β£
x
β£
β₯
1
β
1
x
=
β
1
(
1
x
)
,
x
>
0
β
1
x
=
Ο
+
β
1
(
1
x
)
,
x
<
0
β
1
x
+
β
1
x
=
Ο
2
,
β£
x
β£
β€
1
β
1
x
+
β
1
x
=
Ο
2
,
β£
x
β£
β₯
1
sin
β1
(βx)
cos
β1
(βx)
tan
β1
(βx)
cot
β1
(βx)
csc
β1
x
sec
β1
x
cot
β1
x
cot
β1
x
sin
β1
x+cos
β1
x
csc
β1
x+sec
β1
x
β
Β
=βsin
β1
x,
=Οβcos
β1
x,
=βtan
β1
x,
=Οβcot
β1
x,
=sin
β1
(
x
1
β
),
=cos
β1
(
x
1
β
),
=tan
β1
(
x
1
β
),
=Ο+tan
β1
(
x
1
β
),
=
2
Ο
β
,
=
2
Ο
β
,
β
Β
β£xβ£β€1
β£xβ£β€1
xβR
xβR
β£xβ£β₯1
β£xβ£β₯1
x>0
x<0
β£xβ£β€1
β£xβ£β₯1
β
Now for the more complicated identities. These come handy very often, and can easily be derived using the basic trigonometric identities.
β
1
x
=
β
1
(
1
β
x
2
)
,
x
β₯
0
β
1
x
=
β
1
(
1
β
x
2
)
,
x
β₯
0
β
1
x
=
Ο
β
β
1
(
1
β
x
2
)
,
x
<
0
β
1
x
+
β
1
y
=
β
1
(
x
+
y
1
β
x
y
)
,
x
y
<
1
β
1
x
+
β
1
y
=
Ο
+
β
1
(
x
+
y
1
β
x
y
)
,
x
y
>
1
β
1
x
β
β
1
y
=
β
1
(
x
β
y
1
+
x
y
)
β
1
x
=
β
1
(
x
1
β
x
2
)
,
x
β
(
0
,
1
)
β
1
x
=
β
1
(
1
β
x
2
x
)
,
x
β
(
0
,
1
)
β
1
x
=
β
1
(
x
x
2
+
1
)
,
x
>
0
β
1
x
=
β
1
(
1
x
2
+
1
)
,
x
>
0
sin
β1
x
cos
β1
x
cos
β1
x
tan
β1
x+tan
β1
y
tan
β1
x+tan
β1
y
tan
β1
xβtan
β1
y
sin
β1
x
cos
β1
x
tan
β1
x
tan
β1
x
β
Β
=cos
β1
(
1βx
2
β
),
=sin
β1
(
1βx
2
β
),
=Οβsin
β1
(
1βx
2
β
),
=tan
β1
(
1βxy
x+y
β
),
=Ο+tan
β1
(
1βxy
x+y
β
),
=tan
β1
(
1+xy
xβy
β
)
=tan
β1
(
1βx
2
β
x
β
),
=tan
β1
(
x
1βx
2
β
β
),
=sin
β1
(
x
2
+1
β
x
β
),
=cos
β1
(
x
2
+1
β
1
β
),
β
Β
xβ₯0
xβ₯0
x<0
xy<1
xy>1
xβ(0,1)
xβ(0,1)
x>0
x>0
β
Find the value of
x
x for which
sin
β‘
(
β
1
(
1
+
x
)
)
=
cos
β‘
(
β
1
(
x
)
)
.
sin(cot
β1
(1+x))=cos(tan
β1
(x)).
We have
β
1
(
1
+
x
)
=
β
1
(
1
1
+
(
1
+
x
)
2
)
β
1
x
=
β
1
(
1
x
2
β
1
)
.
cot
β1
(1+x)
tan
β1
x
β
Β
=sin
β1
(
1+(1+x)
2
β
1
β
)
=cos
β1
(
x
2
β1
β
1
β
).
β
Therefore, we have
sin
β‘
(
β
1
(
1
+
x
)
)
=
cos
β‘
(
β
1
(
x
)
)
sin
β‘
(
β
1
(
1
1
+
(
1
+
x
)
2
)
)
=
cos
β‘
(
β
1
(
1
x
2
β
1
)
)
1
1
+
(
1
+
x
)
2
=
1
x
2
+
1
x
2
+
1
=
(
x
+
1
)
2
+
1
x
2
+
2
x
+
1
=
x
2
x
=
β
1
2
.
β‘
sin(cot
β1
(1+x))
sin(sin
β1
(
1+(1+x)
2
β
1
β
))
1+(1+x)
2
β
1
β
x
2
+1
x
2
+2x+1
x
β
Β
=cos(tan
β1
(x))
=cos(cos
β1
(
x
2
β1
β
1
β
))
=
x
2
+1
β
1
β
=(x+1)
2
+1
=x
2
=β
2
1
β
. Β
β‘
β
β
Cite as: Inverse Trigonometric Identities. Brilliant.org. Retrieved 22:45, February 27, 2020, from https://brilliant.org/wiki/inverse-trigonometric-identities/
Get more Brilliant. Sign up