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Inverses of Trigonometric Functions
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13.1 Inverse Trigonometric Relations
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The equation x sin y (1)
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defines a unique value of x for each given angle y. But when x is given, the equation may have no solution or many solutions. For example: if x 2, there is no solution, since the sine of an angle never exceeds 1. 1 If x 2, there are many solutions y 30 , 150 , 390 , 510 , 210 , 330 , . . . . y arcsin x (2)
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In spite of the use of the word arc, (2) is to be interpreted as stating that y is an angle whose sine is x. Similarly we shall write y arccos x if x cos y, y arctan x if x tan y, etc. The notation y sin 1 x, y cos 1 x, etc. (to be read inverse sine of x, inverse cosine of x, etc.) is also used but sin 1 x may be confused with 1/sin x (sin x) 1, so care in writing negative exponents for trigonometric functions is needed.
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13.2 Graphs of the Inverse Trigonometric Relations
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The graph of y arcsin x is the graph of x sin y and differs from the graph of y sin x of Chap. 7 in that the roles of x and y are interchanged. Thus, the graph of y arcsin x is a sine curve drawn on the y axis instead of the x axis. Similarly the graphs of the remaining inverse trigonometric relations are those of the corresponding trigonometric functions, except that the roles of x and y are interchanged. (See Fig. 13.1)
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13.3 Inverse Trigonometric Functions
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It is sometimes necessary to consider the inverse trigonometric relations as functions (i.e., one value of y corresponding to each admissible value of x). To do this, we agree to select one out of the many angles corresponding 1 to the given value of x. For example, when x 1, we agree to select the value y /6, and when x 2 2 , we agree to select the value y /6. This selected value is called the principal value of arcsin x. When only the principal value is called for, we write Arcsin x, Arccos x, etc. Alternative notation for the principal value of the inverses of the trigonometric functions is Sin 1 x, Cos 1 x, Tan 1 x, etc. The portions of the graphs on which the principal values of each of the inverse trigonometric relations lie are shown in Fig. 13.1(a) to ( ) by a heavier line.
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CHAPTER 13 Inverses of Trigonometric Functions
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Fig. 13.1
When x is positive or zero and the inverse function exists, the principal value is defined as that value of y which lies between 0 and 1p inclusive. 2
EXAMPLE 13.1 (a) Arcsin 23>2
p>3 since sin p>3 p>6 since cos p>6
23>2 and 0 < /3 < /2. 23>2 and 0 < /6 < /2.
(b) Arccos 23>2 (c) Arctan 1
/4 since tan /4
1 and 0 < /4 < /2.
When x is negative and the inverse function exists, the principal value is defined as follows:
1 2p 1 2p 1 2p
Arcsin x < 0 < Arccos x < Arctan x < 0
1 2p 1 2p 1 2p
< Arccot x < < Arcsec x Arccsc x < 0
EXAMPLE 13.2 (a) Arcsin A
23>2 B 2 /3
p>3 p>6
Arccot ( 1) Arcsec ( 2/ 3) Arccsc A 22 B
3 /4 5 /6 p>4
(b) Arccos ( 1/2) (c) Arctan A 1> 23 B
CHAPTER 13 Inverses of Trigonometric Functions
Principal-Value Range
Authors vary in defining the principal values of the inverse functions when x is negative. The definitions given are the most convenient for calculus. In many calculus textbooks, the inverse of a trigonometric function is defined as the principal-valued inverse, and no capital letter is used in the notation. This generally causes no problem in a calculus class.
Inverse Function y y 0 y y y y Arcsin x Arccos x y Arctan x Arccot x Arcsec x Arccsc x Principal-Value Range
1 2p
1 2p
1 2p
y y y
1 2p
0<y< 0
1 2p
p, y 2 1 p 2
1 2 p,
13.5 General Values of Inverse Trigonometric Relations
Let y be an inverse trigonometric relation of x. Since the value of a trigonometric relation of y is known, two positions are determined in general for the terminal side of the angle y (see Chap. 2). Let y1 and y2 be angles determined by the two positions of the terminal side. Then the totality of values of y consists of the angles y1 and y2, together with all angles coterminal with them, that is, y1 2n and y2 2n
where n is any positive or negative integer or zero. One of the values y1 or y2 may always be taken as the principal value of the inverse trigonometric function.
EXAMPLE 13.3 Write expressions for the general value of (a) arcsin 1/2, (b) arccos ( 1), and (c) arctan ( 1).
(a) The principal value of arcsin 1/2 is /6, and a second value (not coterminal with the principal value) is 5 /6. The general value of arcsin 1/2 is given by /6 2n 5 /6 2n
where n is any positive or negative integer or zero. (b) The principal value is and there is no other value not coterminal with it. Thus, the general value is given by 2n , where n is a positive or negative integer or zero. (c) The principal value is /4, and a second value (not coterminal with the principal value) is 3 /4. Thus, the general value is given by /4 2n 3 /4 2n