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1. The PrincipalValue Sine Function And Its Inverse Arcsine 
Let f(x) = x^{2}. Clearly dom
( f ) = R.
Consider the “ part ” of f on [0, 1]. Certainly this part isn't the same
function as f,
because its domain is [0, 1], different from that of f.
Let's call it g. We see that dom
( g) is a subset of dom ( f ), and
g(x) = x^{2} = f(x) for all x in dom
( g). The function g
is obtained by restricting dom ( f ) to [0,
1]. We say that g is the
restriction of f to [0, 1].
Consider the function y = sin
x. See Fig. 1.1. We wish to find a function
that's the inverse to sin x or to a
restriction of it.
We saw in Section
3.4 Part 2 that a function is invertible iff it's onetoone. Clearly sin
x isn't onetoone, since
Let's take a look at the inverse of the principalvalue sine
function. For the rest of this section let's abbreviate
“ principalvalue”
as “ PV”. As seen in Section
6.1.1 Part 4, the radian measure x of the
angle is equal to the
Fig. 1.1 
Fig. 1.2

Fig. 1.3

Fig. 1.4

as depicted in Fig. 1.4. 
Graph
Referring to Fig. 1.4 note that:
Recall from Section
3.4 Part 4 that the graph of the inverse f ^{–1} of any invertible function f is the mirror image of
that of f in the line y = x. So the graph of y = arcsin
x is the mirror image of that of y = PV sin x in the line y = x. See Fig.
1.5. In Fig. 1.6, only the graph of y = arcsin x is sketched.
Fig. 1.5 Graph of y = arcsin x is mirror image of that of y 
Fig. 1.6 Graph of y = arcsin x. 
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2. The Principal–Value Cosine Function And Its Inverse Arccosine 
Definition 2.1
as depicted in Fig. 2.3. 
The notation cos^{–1} is also employed for the inverse of
PV cos. We employ the notation arccos because
it's more
suggestive.
We have:
The graph of y = arccos x is the mirror image of that of y = PV cos x in the line y = x. It's sketched in Fig. 2.4.
Fig. 2.1 
Fig. 2.2

Fig. 2.3

Fig. 2.4 Graph of y = arccos x. 
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3. The Principal–Value Tangent Function And Its Inverse Arctangent 
as depicted in Fig. 3.3. 
The graph of y = arctan x is the mirror image of that of y = PV tan x in the line y = x. It's sketched in Fig. 3.4.
Fig. 3.1 
Fig. 3.2

Fig. 3.3

Fig. 3.4 Graph of y = arctan x. 
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4. The Principal–Value Cotangent Function And Its Inverse 
as depicted in Fig. 4.3. 
The graph of y = arccot x is the mirror image of that of y = PV cot x in the line y = x. It's sketched in Fig. 4.4.
Fig. 4.1 
Fig. 4.2

Fig. 4.3

Fig. 4.4 Graph of y = arccot x. 
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5. The Principal–Value Secant Function And Its Inverse Arcsecant 
as depicted in Fig. 5.3. 
The graph of y = arcsec x is the mirror image of that of y = PV sec x in the line y = x. It's sketched in Fig. 5.4.
Fig. 5.1

Fig. 5.2

Fig. 5.3

Fig. 5.4 Graph of y = arcsec x. 
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6. The Principal–Value Cosecant Function And Its Inverse Arccosecant 
Definition 6.1
as depicted in Fig. 6.3. 
The notation csc^{–1} is also utilized for the inverse of
PV csc. We utilize the notation arccsc,
because it's more suggestive.
Notice the removed point 0. This is because csc y = 1/sin y
isn't defined at y = 0, where sin
y = 0.
We have:
The graph of y = arccsc x is the mirror image of that of y = PV csc x in the line y = x. It's sketched in Fig. 6.4.
Fig. 6.1 
Fig. 6.2

Fig. 6.3

Fig. 6.4 Graph of y = arccsc x. 
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7. Expressions Involving The Inverse Trigonometric Functions 
Evaluate arcsin (1/2).
Example 7.2
Simplify tan arcsin x. (That is, express tan arcsin x as an algebraic expression in x.)
Solution 1
EOS
Solution 2
Fig. 7.1 sin y = x/1 = x. 
EOS
it as an algebraic expression in x. In
Solution 2 we label the side opposite to y and the
hypotenuse of the right triangle in
such a way that sin y = x. We choose the simplest labelling: the opposite
side is x and the hypotenuse is 1. We would
obtain the same answer for tan arcsin x if we chose a different labelling, such as
this: the opposite side is 2x and the
hypotenuse is 2. The side adjacent to y is obtained
by the Pythagorean formula.
Problems & Solutions 
1. Evaluate each of the following expressions.
Solution
2. Simplify the expression cos arctan x.
Solution
3. Find:
Solution
4. Show that:
Solution
is established.
5. A careless
mathematics professor asked his calculus class on their final examination to
simplify the expression
sin arccos (2 + x^{2}). What's wrong
with this problem?
Solution
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