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Chapter 4 Section 1 - Definition The Inverse Sine Function...

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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 one-to-one 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 one-to-one. However, if we restrict the domain to the interval [—77/ 2, 77/2], then the restricted function is One- to-one 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 xjaadr-7'7/2-Syé 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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