The Inverse Sine, Cosine, And Tangent Functions Worksheet Template
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6Section 8.1 – The Inverse Sine, Cosine, and Tangent Functions
You learned about inverse functions in both college algebra and precalculus. The main characteristic of inverse functions
is that composing one within the other always equals “x”. In mathematical notation, this is written as follows:
x
7
1
1
1
f f
( )
x
x
and
f
f x
( )
x
f x
( ) 3
x
7
f
( )
x
. For instance, if
, then
. Let’s prove that these are inverse
3
functions:
You may also remember that the domain (x-values) of
1
f x
( )
f
( )
x
is the same as the range (y-values) of
, and
vice versa. Thus, you can get points on one graph just
by interchanging the x and y values from the other.
This results in graphs that are symmetric across the
f x
( ) 3
x
7
line y = x. The graphs of
and
x
7
1
f
( )
x
are shown in the figure to the right.
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Notice that they are mirror images across the line y = x.
f x
( )
Notice also that
contains the points (-2, 1) and
1
f
( )
x
(-1, 4), and that
contains the points
____________ and _____________.
In order to have an inverse, a function must be one-to-one, which means that its graph must pass the Horizontal Line
Test. If a function is not one-to-one, it is usually possibly to restrict its domain to make it one-to-one. Think of the
graphs of the trig functions, such as sine, cosine, and tangent. Are these functions one-to-one? _________ In order to
have inverse trig functions, we must restrict the domains of the functions to make them one-to-one. We will do this as
sin( ) x
x
x
follows: for
, restrict the domain to
, and for tan(x), restrict the domain to
(so we are
2
2
2
2
cos( ) x
only looking at sine and tangent for angles in Quadrants _______ and _______ ). For
, restrict the domain to
0 x
(so we are only looking at cosine for angles in Quadrants _______ and _______ ). These restricted domains
become the _____________________ of the inverse trig functions.
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