Inverse Trig Functions

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Inverse Trig Functions By Richard Gill
Supported in Part by a Grant from VCCS
LearningWare
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Let us begin with a simple question
What is the first pair of inverse functions that
pop into YOUR mind?
This may not be your pair but this is a famous
pair. But something is not quite right with this
pair. Do you know what is wrong?
Congratulations if you guessed that the top
function does not really have an inverse because
it is not 1-1 and therefore, the graph will not
pass the horizontal line test.
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Consider the graph of
Note the two points on the graph and also on the
line y4.
f(2) 4 and f(-2) 4 so what is an inverse
function supposed to do with 4?
By definition, a function cannot generate two
different outputs for the same input, so the sad
truth is that this function, as is, does not have
an inverse.
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So how is it that we arrange for this function to
have an inverse?
We consider only one half of the graph x gt
0. The graph now passes the horizontal line test
and we do have an inverse
Note how each graph reflects across the line y
x onto its inverse.
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A similar restriction on the domain is necessary
to create an inverse function for each trig
function.
Consider the sine function.
You can see right away that the sine function
does not pass the horizontal line test.
But we can come up with a valid inverse function
if we restrict the domain as we did with the
previous function.
How would YOU restrict the domain?
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Take a look at the piece of the graph in the red
frame.
We are going to build the inverse function from
this section of the sine curve because
This section picks up all the outputs of the sine
from 1 to 1.
This section includes the origin. Quadrant I
angles generate the positive ratios and negative
angles in Quadrant IV generate the negative
ratios.
Lets zoom in and look at some key points in this
section.
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I have plotted the special angles on the curve
and the table.
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The new table generates the graph of the inverse.
To get a good look at the graph of the inverse
function, we will turn the tables on the sine
function.
The range of the chosen section of the sine is
-1 ,1 so the domain of the arcsin is -1, 1.
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Note how each point on the original graph gets
reflected onto the graph of the inverse.
etc.
You will see the inverse listed as both
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In the tradition of inverse functions then we
have
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The thing to remember is that for the trig
function the input is the angle and the output is
the ratio, but for the inverse trig function the
input is the ratio and the output is the angle.
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The other inverse trig functions are generated by
using similar restrictions on the domain of the
trig function. Consider the cosine function
What do you think would be a good domain
restriction for the cosine?
Congratulations if you realized that the
restriction we used on the sine is not going to
work on the cosine.
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The chosen section for the cosine is in the red
frame. This section includes all outputs from 1
to 1 and all inputs in the first and second
quadrants.
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The other trig functions require similar
restrictions on their domains in order to
generate an inverse.
yarctan(x)
ytan(x)
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The table below will summarize the parameters we
have so far. Remember, the angle is the input for
a trig function and the ratio is the output. For
the inverse trig functions the ratio is the input
and the angle is the output.
arcsin(x) arccos(x) arctan(x)
Domain
Range
When xlt0, yarcsin(x) will be in which quadrant?
ylt0 in IV
When xlt0, yarccos(x) will be in which quadrant?
ygt0 in II
ylt0 in IV
When xlt0, yarctan(x) will be in which quadrant?
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The graphs give you the big picture concerning
the behavior of the inverse trig functions.
Calculators are helpful with calculations (later
for that). But special triangles can be very
helpful with respect to the basics.
Use the special triangles above to answer the
following. Try to figure it out yourself before
you click.
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OK, lets try a few more. Try them before you peek.
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Negative inputs for the arccos can be a little
tricky.
From the triangle you can see that arccos(1/2)
60 degrees. But negative inputs for the arccos
generate angles in Quadrant II so we have to use
60 degrees as a reference angle in the second
quadrant.
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You should be able to do inverse trig
calculations without a calculator when special
angles from the special triangles are involved.
You should also be able to do inverse trig
calculations without a calculator for quadrantal
angles.
Its not that bad. Quadrantal angles are the
angles between the quadrantsangles like
To solve arccos(-1) for example, you could draw a
quick sketch of the cosine section
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But a lot of people feel comfortable using the
following sketch and the definitions of the trig
ratios.
Good luck getting that answer off of a calculator.
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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 120
degrees? This problem behaved differently because
the first angle, 120 degrees, 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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First, some calculator problems. 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.
Answers and selected complete solutions can be
found after the exercises.
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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.