Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 6: The Trigonometric Functions And Their Inverses – Section 6.2.1: The Inverse Trigonometric
Functions

 

6.2.1
The Inverse Trigonometric Functions

 

 

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1. The Principal-Value Sine Function And Its Inverse Arcsine

 

Let f(x) = x2. 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) = x2 = 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 one-to-one. Clearly sin x isn't one-to-one, since

 

Let's take a look at the inverse of the principal-value sine function. For the rest of this section let's abbreviate
principal-value” 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

 

 

 

Definition 1.1

 

 

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
= PV sin x in line y = x.

 

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

 

Definition 3.1

 

 

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
    Arccotangent

 

Definition 4.1

 

 

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

 

Definition 5.1

 

 

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

 

Example 7.1

 

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.

 

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Problems & Solutions

 

1.  Evaluate each of the following expressions.

 

     

 

Solution

 

 

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2.  Simplify the expression cos arctan x.

 

Solution

 

 

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3.  Find:

 

   

 

Solution

 

 

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4.  Show that:

 

     

 

Solution

 

 

is established.

 

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5.  A careless mathematics professor asked his calculus class on their final examination to simplify the expression
     sin arccos (2 + x2). What's wrong with this problem?

 

Solution

 

 

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