7.6 The Inverse Trigonometric Function

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7.6 The Inverse Trigonometric Function
Objective To find values of the inverse
trigonometric functions.
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The Inverse Trigonometric Function
When does a function have an inverse?
It means that the function is one-to-one. One-to-o
ne means that every x-value is assigned no more
than one y-value AND every y-value is assigned no
more than one x-value.
How do you determine if a function has an inverse?
Use the horizontal line test (HLT).
Lets take a look at the graphs of all
trigonometric functions
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The Inverse Trigonometric Function
Inverse Sine Function
?/2
?/2
Recall that for a function to have an inverse, it
must be a one-to-one function and pass the
Horizontal Line Test.
f(x) sin x does not pass the Horizontal Line
Test
and must be restricted to find its inverse.
sin x has an inverse function on this interval.
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The Inverse Trigonometric Function
The inverse sine function is defined by y
arcsin x if and only if sin y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arcsin x is 1, 1.
The range of y arcsin x is ?/2 , ?/2.
Example 1
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The Inverse Trigonometric Function
The other inverse trig functions are generated by
using similar restrictions on the domain of the
trig function. Consider the cosine function
Inverse Cosine Function
f(x) cos x must be restricted to find its
inverse.
cos x has an inverse function on this interval.
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The Inverse Trigonometric Function
The inverse cosine function is defined by y
arccos x if and only if cos y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arccos x is 1, 1.
The range of y arccos x is 0 , ?.
Example 2
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The Inverse Trigonometric Function
The other trig functions require similar
restrictions on their domains in order to
generate an inverse.
Like the sine function, the domain of the section
of the tangent that generates the arctan is
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The Inverse Trigonometric Function
The inverse tangent function is defined by y
arctan x if and only if tan y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arctan x is (-?,?) .
The range of y arctan x is (?/2 , ?/2).
Example 3
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The Inverse Trigonometric Function
The inverse cotangent function is defined by y
arccot x if and only if cot y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arccot x is (-?,?) .
The range of y arccot x is (0, ?).
Example 4
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The Graph of Inverse Sine
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The Graph of Inverse Cosine
What is the relation between arcsin(x) and
arccos(x) ?
arccos(x) (-1)arcsin(x) ?/2
arcsin(x) arccos(x) ?/2
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The Graph of Inverse Tangent and Cotangent
What is the relation between arctan(x) and
arcot(x) ?
arccot(x) (-1)arctan(x) ?/2
arctan(x) arccot(x) ?/2
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Graphing Utility Graph the following inverse
functions.
Set calculator to radian mode.
a. y arcsin x
b. y arccos x
c. y arctan x
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Graphing Utility Approximate the value of each
expression.
Set calculator to radian mode.
a. cos1 0.75
b. arcsin 0.19
c. arctan 1.32
d. arcsin 2.5
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Composition of Functions
Composition of Functions f(f 1(x)) x
and (f 1(f(x)) x.
Inverse Properties
If 1 ? x ? 1 and ?/2 ? y ? ?/2,
then sin(arcsin x) x and arcsin(sin y) y.
If 1 ? x ? 1 and 0 ? y ? ?, then cos(arccos
x) x and arccos(cos y) y.
If x is a real number and ?/2 lt y lt ?/2,
then tan(arctan x) x and arctan(tan y) y.
If x is a real number and 0 lt y lt ?,
then cot(arccot x) x and arccot(cot y) y.
Example 5 tan(arctan 4) 4
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Example 6
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Example 7
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Example 8
a. sin1(sin (?/2)) ?/2
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Example 9
a. sin1(sin (3?/2)) ?/2
does not lie in the range of the arcsin
function, ?/2 ? y ? ?/2.
However, it is coterminal with
which does lie in the range of the arcsin
function.
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Example 10
Solution
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2
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Finally, we encounter the composition of trig
functions with inverse trig functions. The
following are pretty straightforward
compositions. Try them yourself before you click
to the answer.
Did you suspect from the beginning that this was
the answer because that is the way inverse
functions are SUPPOSED to behave? If so, good
instincts but.
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Consider a slightly different setup
This is also the composition of two inverse
functions but
Did you suspect the answer was going to be 2?/3?
This problem behaved differently because the
first angle, 2?/3, was outside the range of the
arcsin. So use some caution when evaluating the
composition of inverse trig functions.
The remainder of this presentation consists of
practice problems, their answers and a few
complete solutions.
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Find the exact value of each expression without
using a calculator. When your answer is an angle,
express it in radians. Work out the answers
yourself before you click.
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On most calculators, you access the inverse trig
functions by using the 2nd function option on the
corresponding trig functions. The mode button
allows you to choose whether your work will be in
degrees or in radians. You have to stay on top of
this because the answer is not in a format that
tells you which mode you are in.
Use a calculator. For 21-24, express your answers
in radians rounded to the nearest hundredth.
Use a calculator. For 17-20, round to the nearest
tenth of a degree.
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Use a calculator. When your answer is an angle,
express it in radians rounded to the hundredths
place. When your answer is a ratio, round it to
four decimal places, but dont round off until
the very end of the problem.
Answers appear in the following slides.
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Answers for problems 1 9.
Negative ratios for arccos generate angles in
Quadrant II.
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Answers for 17 30.
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Assignment P. 289 1 8, 11 14 (only EXACT
value), 19 21 (only T/F, NO counterexample)