The Inverse Trigonometric Functions
These notes amplify on the book’s treatment of inverse trigonometric
functions and supply some needed practice problems. Please see pages 543–
544 for the graphs of sin

1
x
, cos

1
x
, and tan

1
x
.
1
The Arcsine Function
My sine is
x
; who am I? If
x
is any real number such that

x
 ≤
1, there are
infinitely many possible answers. For example, let
x
= 1
/
2. Then sin
y
=
x
when
y
=
π/
6, 5
π/
6,

7
π/
6,

11
π/
6, 13
π/
6, 17
π/
6,

19
π/
6, and so on.
Note however that as
y
travels from

π/
2 to
π/
2, sin
y
travels from

1 to
1. Since sin
y
is continuous and increasing in the interval

π/
2
≤
y
≤
π/
2,
it follows that for any
x
between

1 and 1 there is exactly one
y
between

π/
2 and
π/
2 such that sin
y
=
x
. We can therefore define a new function
sin

1
as follows:
Definition 1.
If

1
≤
x
≤
1, then sin

1
x
(also known as arcsin
x
) is the
number between

π/
2 and
π/
2 whose sine is equal to
x
.
Comment.
If the function
f
has an inverse, that inverse is generally de
noted by
f

1
. The notation sin

1
x
that has just been introduced is (more
or less) in accord with this general convention. Not quite! The sine func
tion
does not
have an inverse, since given sin
t
it is not possible to recover
t
uniquely.
Maybe we are being too fussy.
But the notation can also be a source
of confusion.
For note that sin
2
x
is a standard abbreviation for (sin
x
)
2
,
and sin
3
x
is a standard abbreviation for (sin
x
)
3
. So should sin

1
x
mean
(sin
x
)

1
? Maybe it should.
But it doesn’t!
The notation arcsin
x
is preferred by many mathematicians.
Unfortu
nately, sin

1
seems to be gaining ground over arcsin, maybe because it fits
the cramped space on calculator keyboards better. There are several other
notations, including “Arc sin,” and “asin.”
1
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Pronunciations vary: sin

1
x
can be pronounced “sine inverse (of)
x
,”
“inverse sine (of)
x
,” or even “arc sine
x
.”
Comment.
You don’t have to know a lot about geography to know the
capital of the country whose capital is Amman.
And you don’t have to
know much about trigonometry to find sin(sin

1
(0
.
123)). Indeed it is clear
that
sin(sin

1
x
) =
x
for all
x
between

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 Spring '09
 James
 Tan, Inverse function, Inverse trigonometric functions

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