539 5.2 Trigonometric Functions The screen in Figure 23(a) shows that csc 90° = 1 and sec1-180°2 = -1 using appropriate reciprocal identities. The third entry uses the reciprocal function key x-1 to evaluate sec1-180°2. Figure 23(b) shows that attempting to find sec 90° by entering 1 cos 90° produces an ERROR message, indicating that the reciprocal is undefined. See Figure 23(c). ■ Reciprocal Identities An identity is an equation that is true for all values of the variables for which all expressions are defined. 21x + 32 = 2x + 6 1x + y22 = x2 + 2xy + y2 Identities Recall the definition of a reciprocal. The reciprocal of a nonzero number a is 1 a . Examples: The reciprocal of 2 is 1 2 , and the reciprocal of 8 11 is 11 8 . There is no reciprocal for 0 because 1 0 is undefined. The definitions of the trigonometric functions given earlier in this section were written so that functions in the same column were reciprocals of each other. Because sin u = y r and csc u = r y , sin u = 1 csc u and csc u = 1 sin u , provided sin u ≠0. Also, cos u and sec u are reciprocals, as are tan u and cot u. The reciprocal identities hold for any angle u that does not lead to a 0 denominator. Reciprocal Identities For all angles u for which both functions are defined, the following identities hold true. sin U = 1 csc U cos U = 1 sec U tan U = 1 cot U csc U = 1 sin U sec U = 1 cos U cot U = 1 tan U CAUTION Be sure not to use the inverse trigonometric function keys to find reciprocal function values. For example, consider the following. cos-11-180°2 ≠1cos1-180°22-1 This is the inverse cosine function, which will be discussed later in the text. This is the reciprocal function, which correctly evaluates sec1-180°2, as seen in Figure 23(a). 1cos1-180°22-1 = 1 cos1-180°2 = sec1-180°2 The reciprocal identities can be written in different forms. For example, sin U = 1 csc U is equivalent to csc U = 1 sin U and 1sin U2 1csc U2 =1. (b) (c) Figure 23 (a)
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