6.2.1 The Inverse Trigonometric Functions
| 6.2.1 |
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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 |
| 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 |
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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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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