607 6.2 The Unit Circle and Circular Functions We have defined the six trigonometric functions in such a way that the domain of each function was a set of angles in standard position. These angles can be measured in degrees or in radians. In advanced courses, such as calculus, it is necessary to modify the trigonometric functions so that their domains consist of real numbers rather than angles. We do this by using the relationship between an angle u and an arc of length s on a circle. 6.2 The Unit Circle and Circular Functions ■ Circular Functions ■ Values of the Circular Functions ■ Determining a Number with a Given Circular Function Value ■ Linear and Angular Speed x y 0 x = cos s y = sin s (x, y) (1, 0) (0, 1) (–1, 0) (0, –1) Arc of length s The unit circle x2 + y2 = 1 u Figure 12 Circular Functions The following functions are defined for any real number s represented by a directed arc on the unit circle. sin s =y cos s =x tan s = y x 1 x 302 csc s = 1 y 1 y 302 sec s = 1 x 1 x 302 cot s = x y 1 y 302 LOOKING AHEAD TO CALCULUS Students planning to study calculus should become very familiar with radian measure. In calculus, the trigonometric or circular functions are always understood to have real number domains. The unit circle is symmetric with respect to the x-axis, the y-axis, and the origin. If a point 1a, b2 lies on the unit circle, so do 1a, -b2, 1-a, b2, and 1-a, -b2. Furthermore, each of these points has a reference arc of equal magnitude. For a point on the unit circle, its reference arc is the shortest arc from the point itself to the nearest point on the x-axis. (This concept is analogous to the reference angle concept.) Using the concept of symmetry makes determining sines and cosines of the real numbers identified in Figure 13* on the next page a relatively simple procedure if we know the coordinates of the points labeled in quadrant I. * The authors thank Professor Marvel Townsend of the University of Florida for her suggestion to include Figure 13. Circular Functions In Figure 12, we start at the point 11, 02 and measure an arc of length s along the circle. If s 70, then the arc is measured in a counterclockwise direction, and if s 60, then the direction is clockwise. (If s = 0, then no arc is measured.) Let the endpoint of this arc be at the point 1x, y2. The circle in Figure 12 is the unit circle — it has center at the origin and radius 1 unit (hence the name unit circle). Recall from algebra that the equation of this circle is x2 + y2 = 1. The unit circle The radian measure of u is related to the arc length s. For u measured in radians and for r and s measured in the same linear units, we know that s = r u. When the radius has measure 1 unit, the formula s = r u becomes s = u. Thus, the trigonometric functions of angle u in radians found by choosing a point 1x, y2 on the unit circle can be rewritten as functions of the arc length s, a real number. When interpreted this way, the trigonometric functions are called circular functions.
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