Unformatted text preview: Definition: The Inverse
Sine Function. 4.1 The Inverse Trigonometric Functions 211 ,4_1 The Inverse Trigonometric Functions We introduced the inverses of the sine, cosine, and tangent functions in Section 1.5.
A discussion of inverse functions in general can be found in Section P.4. In this sec tion we study those functions in more detail and define the inverses of the other
three trigonometric functions. The Inverse Sine Function An inverse of a function is a function that reverses What the function does. If there is , a onetoone correspondence between the elements of the domain and the elements of the range of a function, then the function is invertible. Since y = sinx with do
main (—00, 00) is a periodic function, it is certainly not onetoone. However, if we
restrict the domain to the interval [—77/ 2, 77/2], then the restricted function is One
toone and invertible. Other intervals could be used, but this interval is chosen to
'keep the inverse function as simple as possible. The graph of the sine function with domain [—77/ 2, 77/ 2] is shown in Fig. 4.1(a).
Its range is [—'l, l]. The inverse of this restricted sine function is denoted as
fT1(x) = sin—1(x) (read “inverse sine of x”) or J“1 (x) = arcsin(x) (read “arc sine
of x”). , aya i‘siinﬂtx) Provided 3mm ,5 xjaadr7'7/2Syé 7/2 , ” Figure 4.1 The domain ofy = sin—1(x) is [—1, l] andits range is [—77/2, 77/2]. The graph of
y = sin—1(x) is a reﬂection about the line y = x of the graph of y = sin(x) on
[—7772, 77/2] as shown in Fig. 4.1(b). Depending on the context, sinT1 x might be an angle, a measure of an angle in
degrees or radians, the length of an arc of the unit circle, or simply a real number.
The expression sin‘1 x can be‘read as “the angle whose sine is x” or “the arc length
whose sine is x.” The notation y = arcsin x reminds us that y is the arc length whose
sine is x. For example, arcsin(l) is the arc length/fin [—77/ 2, 77/2] whose sine is 1.
Since we know that sin(77/ 2) = l, we have arcsin(l) = 77/2. We Will assume that
sin‘1 x is a real number unless indicated otherwise. ...
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 Summer '11
 BryanWhite
 Trigonometry

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