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Subsections
The purpose of this lab is to increase your familiarity with the
inverse trigonometric functions.
The inverse trigonometric functions are examples of transcendental
functions. The trigonometric functions are not one to one functions
and hence their inverses are not functions. However, as indicated in
your text, we can restrict the domain of the trigonometric functions
so that the resulting inverse functions are indeed functions.
The
inverse trigonometric functions arise in problems that require finding
angles from side measurements in triangles. They also provide
antiderivatives for a wide variety of functions and hence appear in
solutions to a number of differential equations that arise in
mathematics, engineering and physics.
Some of the Maple commands that you will need to know are as follows:
In order to use the inverse trigonometric functions you must place
arc before the 3 letter symbol for each. For example
arccos(x)is the command for inverse cosine;
arcsin(x)is the command for inverse sine;
arctan(x)is the command for inverse tangent;
arcsec(x)is the command for inverse secant;
arccsc(s)is the command for inverse cosecant;
arccot(x)is the command for inverse cotangent.
Defining x
> x:=0.785
the corresponding value of the inverse tangent can be found as
> evalf(arctan(x));
Remember from previous labs we can take the derivative using either of
the following commands
> diff(arcsec(x),x);
or we can use the D operator with functions
> f:=x>arccsc(x);
> D(f)(x);
We can take the indefinite integral by the following command. Try the following:
> int(arccsc(x),x);
Exercise I:(Definitions, Graphs, Properties)
 1a.
 Graph each of the following trigonometric functions (sin,
cos, tan) and their corresponding inverse function on the same graph. Restrict the domain of each trigonometric function so that the corresponding
inverse function is defined.
 1b.
 Compute the following values.
 a.
 )
 b.
 arcsin(1/2)
 c.
 arctan(a).
 2.
 From 1 above note that the graph of the inverse sine function
is symmetric about the origin. This implies the function is odd.
Show graphically that .
 3.
 Compute and graph the derivative of inverse sine function and the
derivative of inverse cosine function.
 (a)
 Knowing the type of symmetry of the inverse sine function, what
type of symmetry should its derivative have? Does the graph of the
derivative of inverse sine function agree?
 (b)
 How do the graphs of the inverse functions appear to be related?
What transformations can you apply to the graph of the inverse sin
function to obtain the graph of the inverse cos function? What effect
do these transformations have on the graph of the derivative of
inverse sin function? Knowing the derivative of the inverse sin
function, what is a reasonable prediction for the derivative of the
inverse cos function?
Exercise II:(Applications)
 1.
 A billboard to be built parallel to a highway will be 12 m high
with its bottom 4 m above the eye level of the average passing
motorist. How far from the highway should the billbord be placed to
maximize the vertical angle it subtends at the motorist's eyes? Note
that some degree of interpretation is necessary once a numerical
answer has been derived. Can you explain why?
 2.
 Use inverse trigonometric functions to prove that the vertical
angle subtended by a rectangular painting on a wall is greatest when
the painting is hung with its center at the level of the observer's
eye.
 3.
 Evaluate the following integrals and support graphically:
 (a)

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Up: Labs and Projects for
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Christine M Palmer
9/29/1998