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Fig. 68
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In Fig. 68, the line OB ( oh, bee ) is held xed in position, and is called the reference line, or sometimes the initial line. Next, we think of the line OA ( oh, aye ) as free to rotate around the point O, thus generating the angle theta, . The line OA is sometimes called the generating line or, more often, the radius vector. The generating line or radius vector, OA, is free to revolve in either the clockwise (cw) or the counterclockwise (ccw) sense. It has, by general agreement, been decided to call the COUNTERCLOCKWISE sense the POSITIVE sense of rotation. Therefore the opposite sense, the CLOCKWISE direction or sense, is the NEGATIVE sense of rotation. These facts are illustrated in Figs. 69 and 70. Thus, if OA rotates in the POSITIVE direction (ccw), it generates a POSITIVE angle, but if it rotates in the negative direction (cw), it generates a negative angle.
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CHAPTER 5 Sinusoidal Waves. rms Value
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Fig. 69
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Fig. 70
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Any given position of the radius vector OA can be expressed in terms of either a positive angle or a negative angle. An example of this is shown in Fig. 71, wherein the position of OA is the same in both cases.
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Fig. 71
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As you can see in Fig. 71, OA ends up in the SAME POSITION whether we rotate it through a positive angle of 120 degrees or a negative angle of minus 240 degrees. It thus follows that the POSITION of a radius vector OA is not changed if the angle  is increased by any plus or minus integral multiple of 360 degrees; thus, in the example of Fig. 71, the position of OA is not changed if  1208 360n 8 where n is any integer (whole number). In section 5.2, the functions sin , cos , tan , were de ned only for values of  lying in the interval from 08 to 908. In this section, however, we ve just been discussing values of  outside the range of 08 to 908, and this naturally brings up the question, How are the functions sin , cos , tan  to be handled if the angle  lies outside the range of zero degrees to 90 degrees . For example, what meaning is to be given to terms such as sin 1208, cos 3558, tan 4858, and so on In order to give meaning to such terms, the following system has been adopted, and is in universal use today. We make use of the standard x and y rectangular coordinate system, with the four quadrants numbered as shown in Fig. 72. We then let the angle  be generated by rotating the radius vector around the origin, as shown in Fig. 73. It should be emphasized that the four quadrants will always be numbered and referred to as shown in Fig. 72. Next, in Fig. 73, A represents the LENGTH of the radius vector, it being de ned that A is always given as a POSITIVE value. The reference line, from which all angles will be measured, will be the positive side of the x-axis. Positive angles are generated by rotating A in the ccw sense, and negative angles are generated when A is rotated in the cw sense (as in Figs. 69 and 70 above). Now, with all the above in mind, the procedure for nding the values of the trigonometric functions of ANY ANGLE  is as follows. Begin by drawing a line from the tip of the radius vector perpendicular to the x-axis. Be careful to note that this line is always drawn from the tip of the radius vector to the x-axis, regardless of what quadrant the radius vector might be in. Drawing this perpendicular line
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