4.5
INVERSE CIRCULAR FUNCTIONS
From unit 2, we learned that any relation has an inverse (relation), implying that all functions have
inverses. We also learned that for an inverse of a function to be in itself a function, the function must
me onetoone or injective.
Recall
:
A function
is 11 if and only if for any
and
in the domain of
, if
, then
;
or if the graph of
satisfies the conditions the horizontal line test.
TIME TO THINK!
Are circular functions onetoone? Prove or disprove using the definition of 11 functions.
Graphically, it is obvious that all circular functions are not onetoone.
Now, let us look at the inverses of each circular function
Definition 4.5.1
Let
be the sine function defined by
. Then the inverse sine relation or
arcsine
, denoted by
arcsin
, is defined by
where
Example 4.5.1
Since arcsine is not a function, an
arcsin
x
has several “values”, that is
TIME TO THINK!
Give a generalization for the “values” of the following:
In spite of the use of the word “
arc
”,
arcsin
is to be interpreted as stating that “
is an arclength whose
sine is
”. Similarly, we will be using this prefix for the inverse relations of the other circular functions.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Figure 1. The Graph of the Inverse Sine Relation
Definition 4.5.2
Let
be the sine function defined by
. Then the inverse sine relation or
arccossine
, denoted
by
arccos
, is defined by
where
TIME TO THINK!
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 dikopaalam
 Math, Inverse function, Inverse trigonometric functions, arctan function, Arccosecant Function, Arccot Function

Click to edit the document details