PartialTrigonometr
Inverse Trigonometric
Inverse
ic Functions
Inverse Trigonometr
Functionsic Functions
Although the three basic trigonometric functions— -sin. cos. and tan—--do not
have
inversethe
functions.
cos.
and tan—do
tan—--do
Although
do have what
not
the
three basic
basic
they
can be called
trigonometric
‘partia1
inverses”.
functions—
-sin. cos,
Although
three
trigonometric
functions—sin,
and
not
inverse
functions.
do have
have what
These
what
partialfunctions,
they
can be
be and
inverses are
‘partia1 inverses”.
called
inverses”.
arcsine,
have inverse
called
arccosine.
inverse
arct.angent.
have
they
do
can
called
“partial
These
partial
WeJ1partial
and arctangent.
chapter
begin this
with
inverses
review of
are called
calleda arcsine,
the definition
arcsine,
arccosine.
of the most famous
arct.angent.
These
inverses
are
arccosine,
and
example
of a partial
inversewith
function,
WeJ1 begin
the of
function.
chapter
square-root
begin
with
this chapter
review
see
of the
the definition
definition
of the
theThen
most
famous
mostwe’ll
famous
We’ll
this
aa review
of
that
arcsine,
arccosine,
and function,
Thenmanner.
arctangent
we’ll see
see
example
ofaa partial
be definedfunction.
partial
in a similar
inverse
function,
thecan
function.
square-root
example
of
inverse
the
square-root
Then
we’ll
arcsine, arccosine,
arccosine, and
and arctangent
arctangent can
can be
be defined
defined in
in aa similar
similar manner.
that arcsine,
manner.
that
Square-root
Square-root
The graph of x2 is shown below. It does not pass the horizontal line test.
Square-root
Theis,graph
graph
shown below.
That
thereof
the horizontal
line than
lines that
intersect
of x
2 in more
ofare
does not
not the
below.
It does
horizontal
passgraph
test.
The
xx22 horizontal
isis shown
It
pass
the
line
test.
2
that intersect
one
intersect
tells
us that lines
Thisare
2 is not
x
Thatpoint.
the
horizontal
is, there
there
are
that
linesfunction
2meaning
x
in more than
That
is,
horizontal
that
theone-to-one,
graph of x
than
2
there
are two
different
we can
one point.
point.
for themeaning
tells us
usnumbers
that the
as inputs
This
function
2 use
functionthat
thethat
2
x
is not
not
one-to-one,
that
one-to-one,
one
This
tells
that
function
xx
is
(_3)2
(3)2.
that For
we can
can use
that
use as
for
the function
as inputs
inputs= for
in the sanie
therewill
output.
function x
22
x
example,
are result
two different
different
9 the
numbers
di↵erent
there
are
two
numbers
that
we
(_3)2
(3)2.
that will
will result
the same
result in
in the
For example,
sanie output.
output. For
= 9
9 = (3)2 .
example, (−3)
=
( 3)2 =
that
4
c
A function
function has
an inverse
inverse only
has an
only if
if it
it is
is both
A
both one-to-one
one-to-one and
and onto.
onto. (Remember
(Remember
that
a
function
is
onto
if
the
the
target
function
A
of
range
function
an
is
the
inverse
has
of
the
both
only
if
it
is
one-to-one
and
onto.
(Remember
that a function is onto if the target of the function is the range of
the func
funcx22 : is
tion.)
IR
Since
is
the
R
—*
isn’t
one-to-one,
of
the
it
has
iio
inverse.
However,
that
a
function
onto
if
the
we
can
the
target
function
of
range
R→
!R
R isn’t one-to-one, it has no inverse. However, wefunc
tion.) Since x : R
can
x2 : R of
tion.)
IR
Since
—*
isn’t
one-to-one,
restrict
it
has
domain
iio
inverse.
2
However,
the
2
x
nonnegative
to
the
set
of
we can
restrict the domain of x to the set of nonnegative numbers,
numbers, and
and the
the resulting
resulting
2
restrict
domain
of
functionthe
one-to-one. x
is one-to-one.
Wetocan
below
can
see
graph
:
2
x
2
[0,
the see
set this
of nonnegative
numbers,
resulting
1) —÷
! IRR
R
function
is
We
this
below as
as the
the
graph of
ofand
x :the
[0, oc)
∞)
→
passes
the
horizontal
line
test.
There
aren’t
any
horizontal
function
lines
one-to-one.
in
the
plane
is
We
below
can
see
this
as
the
graph
of
:
2
x
oc)
IR
[0,
—÷
passes the horizontal line test. There aren’t any horizontal lines in the plane
that
intersect
graph
the graph
one
intest.
moreThere
than
passes
the horizontal
linein
any horizontal lines in the plane
that
intersect
the
more
than aren’t
one point.
point.
that intersect the graph in more than one point.
c.
JL1
:x
2
[o,°o— 1f
:[o,°o— 1f
2
x
264
294
264
264
c.
4
c
JL1
2 —* —*
TheThe
range
of of
thethe
function
2
x
cc)
JR ∞)
the
is is
setset
numbers
that
range
function
2: [0,
x
: [0,
JR
the
of
numbers
that that
The
range
of
the
function
xcc)
→!
R
isof
the
set
of
numbers
2: [0,
2 [0,
The
range
of
the
function
x
:
1)
R
is
the
set
of
numbers
that
The
range
of
the
function
x
:
[0,
1)
!
R
is
the
set
of
numbers
that
areare
outputs
ofoutputs
thethe
function.
It’sIt’s
thethe
set
ofthe
numbers
that
appear
outputs
as appear
thethe
of
function.
set
of
numbers
yy
that
appear
as
are
of
the
function.
It’s
set
of
numbers
that
as
the
yare
outputs
of of
thethe
function.
It’s
the
setset
of of
numbers
that
appear
as as
thethe
y- youtputs
function.
It’s
numbers
appear
coordinates
ofare
the
points
in
the
graph.
It’s
thethe
setset
cc).
Now
insure
[0, [0,
coordinates
tothat
of
the
points
in
the
graph.
It’s
the
cc).
Now
insure
to
coordinates
of
the
points
in
the
graph.
It’s
the
set
[0,
∞).
coordinates
of of
thethe
points
in
the
graph.
It’sIt’s
thethe
setset
[0, [0,
1).
coordinates
points
in
graph.
1).with
2 thethe
that
we we
have
an an
onto
function,
we’ll
replace
target
of
that
our
function
have
onto
function,
we’ll
replace
the
target
of
our
function
with
We’ll
replace
the
target
of
x
∞)
→
R
with
its
range,
[0,
∞).
The
2: [0,
2
We’ll
replace
the
target
of
x
:
[0,
1)
!
R
with
its
range,
[0,
1).
The
We’ll
replace
the
target
of
x
:
[0,
1)
!
R
with
its
range,
[0,
1).
The
its its
range.
The
result
a
function
that
is
onto
as
well
as
one-to-one
range.
The
result
a
function
that
is as
onto
asaswell
as one-to-one
result
is
a
function
that
is
onto
well
one-to-one
result
is ais function
that
is onto
as as
well
as as
one-to-one
result
a function
that
is onto
well
one-to-one
2 [0, [0,
[0, [0,
cc)x
cc)
cc)
cc)
∞)
∞)
x:2 [0,
1)→
![0,!
[0,
1)1)
x: 2[0,
: [0,
1)
[0,
c.
JL1
0
x:L,oo’)
x:L,oo’)
T
0
Any
function
that
is isone-to-one
and
onto
has
Therefore,
the
Any
function
that
one-to-one
andand
onto
hasan
aninverse.
inverse.
Therefore,
thethe
Any
function
that
is one-to-one
onto
has
an
inverse.
Therefore,
2
function
x
:
[0,
∞)
→
[0,
∞)
has
an
inverse.
We
named
this
inverse
function
2
2
Any
function
that
and
onto
hasan
an
inverse.
Therefore,
Any
function
the
that
is[0,: one-to-one
and
onto
has
an
inverse.
Therefore,
function
xis : one-to-one
1)1)
!!
[0,
1)
has
an
inverse.
WeWe
named
thisthis
inverse
the function
function
[0,
[0,
1)
has
inverse.
named
inverse
function
√
p
p[0, ∞) →x [0,
x
:
∞).
To
see
the
graph
of
the
inverse
function,
we
just
have
function
: x
2x
x
[0,: :x
cc)
cc)
has1).
—*1)
[0,
an
inverse.
We
named
this
function
inverse
function
2
[0,
has
—*
We
named
this
[0,: cc)
1)
![0,
[0,cc)
1).
Toan
seeinverse.
thethe
graph
of of
the
inverse
function,
wewe
just
have
inverse
function
[0,
!
[0,
To
see
graph
the
inverse
function,
just
have
to
flip
the
graph
of
our
original
function
over
the
y
=
x
line.
/:/:[0, [0,
cc)cc)
cc).
Tograph
—+to
[0,flip
seeof
the
graph
of the
inverse
function,
we
just
have
To
—+
[0,
see
the
graph
offunction
to
flip
thecc).
graph
our
original
over
thethe
y=
line.
the
inverse
function,
we
have
the
of
our
original
function
over
yx
=
xjust
line.
flipflip
to to
thethe
graph
of
our
original
function
over
the
graph of our original function over the
y =y x= line.
x line.
0
0
T
Because
the
x2x:2 x[0,
∞)
∞)
only
part
ofofthe
function
x2x2 x2
Because
thefunction
function
: 2[0,
1)→
![0,
[0,
1)
is
only
part
the
function
Because
the
function
: [0,
1)
!
[0,
1)
is
only
part
of
the
function
√is
p
p the inverse of this smaller
with
itsitsimplied
ofofR,of
because
x xis
with
implied
domain
R,and
andand
because
isx the
inverse
of of
this
smaller
with
its
implied
domain
R,
because
is the
inverse
this
smaller
√pdomain
ppartial inverse of x2 . 2 2
x
a
part,
we
call
part,
wewe
callcall x ax partial
inverse
of of
x .x .
part,
a 2partial
inverse
√p p[0,cc) to
What
it means
: [O,cc)
formeans
2x
x
[O,cc)
What
—÷
itWhat
means
andand
: and
[O,cc)
:for[O,cc)
for
2
[O,cc)
be be
—÷
: [O,cc)
[0,cc)
to
it
x
:
∞)
→
[0,
∞)
x—
:—
[0,
∞)
→
[0,
∞)
2 [0,
2
What
it
means
for
x
:
[0,
1)
!
[0,
1)
and
x
:
[0,
1)
!
[0,
1)to
tobeto
be be
What
it
means
for
x
:
[0,
1)
!
[0,
1)
and
x
:
[0,
1)
!
[0,
1)
inverses
isinverses
that
they
reverse
inverses
each
other’s
assignments.
is that
they
reverse
For
example,
because
each
other’s
assignments.
For
example,
because
is
that
they
reverse
each
other’s
assignments.
For
example,
because
√p
√
inverses
is that
they
reverse
each
other’s
assignments.
example,
because
inverses
is that
they
reverse
each
other’s
assignments.
For
example,
because
52 52
32 For
32
p
p
p3 that
2
/i
= 25,
we
know
that
= 5.
Because,
/i
= 25,
know
that
= 3
know
The
= 5.
Because,
= that
9•
= we
3 we
know
that
The
= 9•
525we
=
we
know
that
25
=
5.
Because,
9
=
we
know
3
=
The
2 25,
2 9.
2
2
wewe
know
that
25 25
= 5.
Because,
know
that
3 3= 9.
The
5= 25,
= more
25,
know
that
= 5.
Because, 9 =9 3=we
3 we
know
that
= 9.
The
chart
below
shows
examples
of the
chart
below
dual
shows
relationship
between
these
twotwo
more
examples
of the
dual
relationship
between
these
chart
below
shows
more
examples
of
the
dual
relationship
between
these
two
chart
below
shows
more
examples
of the
dual
relationship
between
these
twotwo
chart
below
shows
more
examples
of the
dual
relationship
between
these
inverse
functions.
inverse
functions.
inverse
functions.
inverse
functions.
inverse
functions.
265 265
295
265 265
√
x2
√
02 = 0
√
12 = 1
√
22 = 4
√
32 = 9
Graphs of sine and cosine
42 = 16
52 = 25
√
√
x
0=0
1=1
4=2
9=3
16 = 4
25 = 5
I
Arcsine
Sine is not one-to-one. Its graph fails the horizontal line test.
sin(e)
‘2.
Zir
296
Identities for sine and cosine
An identity is an equation in one variable that is true for every possible
value of the variable. For example, x + x
2x is an identity because it’s
always true. It does’t matter whether x equals 1, or 5, or
it’s always true
—
There isis however
however aa segment
segment of
of the
the graph
graph of
of sin
sin that
that satisfies
satisfies the
the horizontal
horizontal
There
line test.
test.
line
5mn
“2.
2
Notice that
that the
the segment
segment of
of the
the graph
graph highlighted
highlighted above
above stretches
stretches between
Notice
π
π
and on the x-axis, so it is the graph of sin with its domain
−—
2 and 2 on the x-axis, so it is the graph of sin with its domain restricted
π π
π π —÷ R. Because this graph passes the
to [−
It’s the
the graph
graph of
of sin
sin :: [−
to
]. It’s
[—i,
[—.,
2 , 2].
2 , 2]] → R. Because this graph
π π
horizontal
one
test,
it
is
one-to-one.
horizontal line test, sin : [− 2 , 2 ] → R is one-to-one.
2
/2
‘2
π π —* IR is the set of numbers that appear
The range
range of
of sin
sin :: [−
The
appear as the
[—i,
2 , 2]] → R is the set of numbers
y-coordinates of
of points
points in
in its
its graph.
graph. This
This set
set is
is [−1,
[—1, 1].
1]. Replacing
Replacing the target
y-coordinates
of our
our function
function with
with its
it’srange
range gives
gives us
us aa function
function that
that isis onto,
onto, in
in addition
addition to
of
being one-to-one.
one-to-one. That
That is,
is,
being
h π πi
sin:
sin : − , 1 → [-1,1]
[−1, 1]
2 2
one-to-one and
and onto,
onto, and
and thus
thus has
has an
an inverse.
inverse. We
We could
could write this inverse
isis one-to-one
as
as
h π πi
−1
:
1
sin
[—1,1]
sin
: [−1,
1] → − ,
2 2
[-i,
[—i,
267
297
=:
=:
14
14
but because this partial inverse of sin is an important function in math, it has
been given a special name. It’s called the arcsine function, and it’s written
as
h π πi
arcsin : [−1, 1] → − ,
2 2
To find the graph of arcsin, we flip the graph of sin : [− π2 , π2 ] → [−1, 1] over
the y = x line.
The chart on the next page demonstrates some values of the arcsine function. Because arcsine is the inverse of sin : [− π2 , π2 ] → [−1, 1], it reverses the
assignments of sin : [− π2 , π2 ] → [−1, 1].
298
sin(x)
arcsin(x)
sin(− π2 ) = −1
arcsin(−1) = − π2
sin(− π3 )
√
=−
√
3
2
arcsin(−
3
2 )
= − π3
sin(− π4 ) = − √12
arcsin(− √12 ) = − π4
sin(− π6 ) = − 12
arcsin(− 12 ) = − π6
sin(0) = 0
arcsin(0) = 0
sin( π6 ) =
1
2
arcsin( 12 ) =
√1
2
arcsin( √12 ) =
sin( π4 ) =
sin( π3 )
√
=
√
3
2
arcsin(
sin( π2 ) = 1
3
2 )
=
arcsin(1) =
π
6
π
4
π
3
π
2
Arccosine
Cosine is also not one-to-one. Its graph fails the horizontal line test.
299
Graphs cos(e)
of sine and cosine
I
The below segment of the cosine graph does satisfy the horizontal line test.
sin(e)
Zir
‘2.
I
1
Identities for sine and cosine
An identity is an equation in one variable that is true for every possible
value
The segment
stretchesFor
between
0 and
It’s the
graph it’s
of
of the variable.
example,
x +πx on the
2x isx-axis.
an identity
because
alwaysrestricted
cosine
to thematter
domainwhether
[0, π], the
graph1,oforcos
: [0, π] →
Because
true. It does’t
x equals
5, or
it’sR.
always
true
its
graph
passes the horizontal line test, cos : [0, π] → R is one-to-one.
that
x+x=2x.
The remainder of this chapter is an assortment of important identities for
the functions sine and cosine.
—
Lemma
(7). (The Pythagorean identity) For any number 8,
cos(e)
2 +sin(O)
cos(8)
2
=
1
Proof: The equation for the unit circle is 2
+Cos:[O,7r]
x
y = 1. Since (cos(O), sin(8))
is a point on the unit circle, it is a solution of this equation. That is,
—
2
2 + sin(O)
cos(8)
=
1
S
209
300
To make cos : [0, π] → R into an onto function, we can replace its target
with its range. To do this, we just need to identify the range. It’s the set
of numbers that appear as y-coordinates of points in the graph. It’s the set
[−1, 1]. Thus, cos : [0, π] → [−1, 1] is onto as well as one-to-one. Therefore,
it has an inverse function that’s called the arccosine function:
arccos : [−1, 1] → [0, π]
It’s graph is obtained by flipping the graph of cos : [0, π] → [−1, 1] over the
y = x line.
ir
av-ccos: [i,i]
__;
[o,1TJ
The arccosine function is a partial inverse of the cosine function. The chart
on the next page shows some values of arccosine.
301
cos(x)
arccos(x)
cos(0) = 1
arccos(1) = 0
√
=
3
2
cos( π4 ) =
√
3
2 )
=
π
6
√1
2
arccos( √12 ) =
π
4
1
2
arccos( 12 ) =
π
3
cos( π2 ) = 0
arccos(0) =
π
2
1
cos( 2π
3 ) = −2
arccos(− 12 ) =
√1
cos( 3π
4 ) =− 2
arccos(− √12 ) =
cos( π6 )
cos( π3 ) =
cos( 5π
6 )
arccos(
√
=−
3
2
√
arccos(−
cos(π) = −1
3
2 )
=
2π
3
3π
4
5π
6
arccos(−1) = π
Arctangent
Tangent is not one-to-one, but the segment of its graph between − π2 and
on the x-axis does pass the horizontal line test.
302
π
2
II
/
I
17•
/
I
/
I
I
I
I
‘I
Therefore, the restricted function tan : (− π2 , π2 ) → R is one-to-one.
N)
N)
303
I
I
1’
-4
1’
-4
It’s also onto, because any real number is the y-coordinate of some point
of its graph. Therefore, tan : (− π2 , π2 ) → R has an inverse function, the
arctangent function
π π
arctan : R → − ,
2 2
π π
Its graph is the graph of tan : (− 2 , 2 ) → R flipped over the y = x line.
The chart below shows the dual relationship between tangent and its partial
inverse, arctangent.
tan(x)
arctan(x)
√
tan(− π3 ) = − 3
√
arctan(− 3) = − π3
tan(− π4 ) = −1
arctan(−1) = − π4
tan(− π6 ) = − √13
arctan(− √13 ) = − π6
tan(0) = 0
arctan(0) = 0
tan( π6 ) =
√1
3
arctan( √13 ) =
tan( π4 ) = 1
tan( π3 ) =
√
arctan(1) =
π
6
π
4
√
arctan( 3) =
3
304
π
3
Exercises
For #1-3, write the values as numbers that do not involve the letters cos,
sin, or tan.
p
p
1.) arcsin( p12 )
2.) arccos( 23 )
3.) arctan( 3)
⇡
4.) sin( 10
)=
⇡
5.) cos( 12
)=
⇡
tan( 10
)
6.)
p
5 1
4 .
p
1+p 3
.
2 2
q
= 1
What does arcsin
What does arccos
p2 .
5
p
5 1
4
p
1+p 3
2 2
What does arctan
equal?
equal?
⇣q
1
For #7-9, match the functions with their graphs.
7.) arcsin(x)
A.)
p2
5
⌘
equal?
8.) arccos(x)
B.)
9.) arctan(x)
C.)
av-ccos: [i,i]
__;
=:
N)
14
[o,1TJ
10.) Write the vector (2, 4) in polar coordinates.
✓
◆✓ ◆
1 0
4
11.) Find the vector
.
2 5
3
305
I
1’
-4
ir
Match the functions with their graphs.
12.) arctan(x)
13.) arctan(x + π2 )
15.) arctan(x − π2 )
16.) arctan(x) +
π
2
19.) − arctan(x)
20.) arctan(−x)
B.)
C.)
18.) arctan(x) −
A.)
π
2
14.) 2 arctan(x)
17.)
1
2
arctan(x)
o
r
‘tr
0
e7r
14
-ir
Ar
2
D.)
E.)
F.)
‘tr
2.
Ar
-IT
a
I
G.)
H.)
2
I.)
o
306
0
Find the set of solutions of the following equations in one variable.
21.) x2 = 25
22.) (x − 2)2 = 9
23.) log3 (x + 1) = 4
2
24.) ex = ex+2
307