609 6.2 The Unit Circle and Circular Functions For tan s, defined as y x , x must not equal 0. The only way x can equal 0 is when the arc length s is p 2 , - p 2 , 3p 2 , - 3p 2 , and so on. To avoid a 0 denominator, the domain of the tangent function must be restricted to those values of s that satisfy s 3 12n +12 P 2 , where n is any integer. The definition of secant also has x in the denominator, so the domain of secant is the same as the domain of tangent. Both cotangent and cosecant are defined with a denominator of y. To guarantee that y ≠0, the domain of these functions must be the set of all values of s that satisfy s 3nP, where n is any integer. Domains of the Circular Functions The domains of the circular functions are as follows. Sine and Cosine Functions: 1 −H, H2 Tangent and Secant Functions: es∣ s 3 12n +12 P 2 , where n is any integerf Cotangent and Cosecant Functions: 5 s∣ s 3nP, where n is any integer6 Values of the Circular Functions The circular functions of real numbers correspond to the trigonometric functions of angles measured in radians. Let us assume that angle u is in standard position, superimposed on the unit circle. See Figure 14. Suppose that u is the radian measure of this angle. Using the arc length formula s = r u with r = 1, we have s = u. Thus, the length of the intercepted arc is the real number that corresponds to the radian measure of u. We use the trigonometric function definitions to obtain the following. sin u = y r = y 1 = y = sin s, cos u = x r = x 1 = x = cos s, and so on. As shown here, the trigonometric functions and the circular functions lead to the same function values, provided that we think of the angles as being in radian measure. This leads to the following important result. x y (0, 1) (1, 0) (–1, 0) (0, –1) x2 + y2 = 1 r = 1 x y s = U (cos s, sin s) = (x, y) 0 U Figure 14 Evaluating a Circular Function Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations. Calculators must be in radian mode when they are used to find circular function values.
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