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Trigonometry |
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Inverse Trigonometric Functions or
Arc-functions and their Graphs |
Inverse
functions
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The
arc-sine function and the arc-cosine function |
The
arc-sine function |
The
arc-cosine function |
The
graph of the
arc-sine
function and the arc-cosine
function |
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Inverse Trigonometric Functions or
Arc-functions and their Graphs |
Inverse
functions
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The inverse function, usually written
f -1, is the function whose
domain and the range are respectively the range and domain of a given function
f, that is |
f
-1(x)
= y if and only
if f (y)
= x . |
Thus, the
composition of the inverse function and the given function returns
x, which is called the
identity function,
i.e., |
f
-1(f
(x))
= x and
f (
f
-1(x))
= x. |
The inverse of a function undoes the procedure
(or function) of the given function. |
A pair of inverse functions is in
inverse relation. |
Example: If
given
f (x)
= log2 x
then f -1(x)
= 2x
since, |
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The
arc-sine function and the arc-cosine function |
- The
arc-sine function
y
= sin-1x
or y
= arcsin x
is the inverse of the sine function, so that its value for
any
argument is an arc (angle) whose sine equals the given argument. |
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That
is, y
= sin-1x
if and only if x
= sin y.
For
example, |
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Thus, the arc-sine
function is defined for arguments between -1
and 1, and its principal
values are by
convention taken to be those between -p/2
and p/2. |
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- The
arc-cosine function
y
= cos-1x
or y
= arccos x
is the inverse of the cosine function, so that its
value for any
argument is an arc (angle) whose cosine equals the given
argument. |
That
is, y
= cos-1x
if and only if x
= cos y.
For
example, |
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Thus, the arc-cosine
function is defined for arguments between -1
and 1, and its principal
values are by
convention taken to be those between 0 and p. |
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The
graph of the
arc-sine
function and the arc-cosine
function |
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Functions
contents D |
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