Algebra and Trig. I
4.7 – Inverse Trigonometric Functions
The graph of
If we look at the graph of
we can see that if you draw a
horizontal line between-π and π that the sine function is not 1-1
and therefore does not have an inverse.
However if we restrict our region to
and then we can see that
the sine function is 1-1 and therefore the inverse does exist.
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The inverse of the restricted sine function is called the inverse
sine function.
The inverse sine function denoted
, is the restricted sine function.
. Thus:
where
and
equals the inverse sine at x”
. We read
as “y
The following are points on the restricted
. Note if we
reverse the order of the points we can see the points of
x
0
y=sinx
0
0
x
y=
x
0
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Also note that the inverse of a function is the graph of the function
reflected across the line y=x.
Let's see the graph of y = sin x first and then derive the curve of
y = arcsin x.
If we reflect the indicated portion of y = sin x through the line y = x, we
obtain the graph of y = arcsin x:
What you see is what you get. The graph does not extend beyond the
indicated boundaries of x and y.
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The domain (the possible x-values) of arcsin x is
The range (of y-values for the graph) for arcsin x is
Exact values of
angle in the interval
can be found by thinking of
whose sine is x.
as the
For example:
“the angle where sine is -1 is
”
“the angle where sine is 1 is ”
Finding exact values of
1. Let
2. Rewrite
as
3. Use the exact values of sin(x) to find the value of θ in
that satisfies sin(θ)
Example – Find the exact value of
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The graph of
If we restrict our region to and then we can see that the
cosine function is 1-1 and therefore the inverse does exist.
The inverse cosine function denoted
inverse of the restricted cosine function.
Thus:
where
and
equals the inverse cosine at x”
, is the
.
as “y
. We read
The following are points on the restricted
. Note if we
reverse the order of the points we can see the points of
x
0
π
y=cosx
1
-1
x
y=
x
1
-1
0
π
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Let's first recall the graph of y = cos x so we can see where the graph
of y = arccos x comes from.
We now choose the portion of this graph from x = 0 to x = π.
The graph of the inverse of cosine x is found by reflecting the graph
of cos x through the line y = x.
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We now reflect every point on this portion of the cos x curve through
the line y = x.
The result is the graph y = arccos x:
That's it for the graph - it does not extend beyond what you see here.
(If it did, there would be multiple values of y for each value of x and
then we would no longer have a function.)
The domain (the possible x-values) of arccos x is -1 ≤ x ≤ 1
The range (of y-values for the graph) for arccos x is 0 ≤ arccos x ≤ π
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Finding exact values of
1. Let
2. Rewrite
as
3. Use the exact values of cos(x) to find the value of θ in
that satisfies cos(θ)
Example – Find the exact value of
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The graph of
If we restrict our region to
and then we can see that the
tangent function is 1-1 and therefore the inverse does exist.
The inverse tangent function denoted
inverse of the restricted tangent function.
Thus:
where
and
equals the inverse tangent at x”
. We read
, is the
.
as “y
Let's first recall the graph of y = tan x so we can see where the graph
of y = arctan x comes from.
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Reflecting this portion of the graph in the line y = x, we obtain the
graph of y = arctan x:
This time the graph does extend beyond what you see, in both the
negative and positive directions of x.
The domain (the possible x-values) of arctan x is All values of x
The range (of y-values for the graph) for arctan x is
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Finding exact values of
1. Let
2. Rewrite
as
3. Use the exact values of tan(x) to find the value of θ in
that satisfies tan(θ)
Example – Find the exact value of
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Composition of Functions Involving Inverse Trigonometric
Functions
Inverse Properties
The Sine Function and its Inverse
for every
for every
The Cosine Function and its Inverse
for every
for every
The Tangent Function and its Inverse
for every
for every
Example – Find the exact value, if possible:
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Hannah Province – Mathematics Department – Southwest Tennessee Community College