is    inscribed    within    the    circle.    Coordinates measured from 0 along the x axis to thc right are positive; coordinates measured from  0 along the x  axis  to  the  left  are  negative.  Coordinates measured  along  the  y axis  from  0  upward  are positive;  coordinates  measured  along  the  y  axis from  0  downward  are  negative. Angles are generated by the motion of a point P  counterclockwise  along  the  circumference  of  the circle. The initial leg of any angle is the positive leg of the x axis. The other leg is the radius r, at the end of which the point P is located; this radius always has a value of 1. The unit radius (r = OC) is subdivided into 10 equal parts, so the value of each  of  the  10  subdivisions  shown  is  0.1. For  any  angle, the   point   P   has   three coordinates:  the  x  coordinate,  the  y  coordinate, and the r coordinate (which always has a value of 1 in this case). The functions of any angle are, collectively,  various  ratios  that  prevail  between these  coordinates. The  ratio  between  y  and  r  (that  is,  y/r)  is called  the  sine  of  an  angle.  In  figure  1-21,  AP seems  to  measure  about  0.7  of  y;  therefore,  the sine 8, which is equal to 45° in this case, would seem to be 0.7/1, or about 0.7. Actually, the sine of 45° is 0.70711. Graphically, the sine is indicated in figure 1-21 by the line AP, which measures 0.7 to  the  scale  of  the  drawing. The  ratio  between  x  and  r  (that  is,  x/r)  is called the cosine  of the angle. You can see that for 45°, x and y are equal, and the fact that they are  can  be  proven  geometrically.  Therefore,  the cosine  of  45°  is  the  same  as  the  sine  of  45°,  or 0.70711.    Graphically,   the  length  of  line  OA represents the cosine of angle  0 when the radius (r) is equal to 1. The  ratio  between  y  and  x  (that  is,  y/x)  is known as the tangent of an angle. Since y and x for an angle of 45° are equal, it follows that the tangent of an angle of 45° equals 1. The tangent is also indicated graphically by the line BC, drawn tangent  to  the  circle  at  C  and  intersecting  the extended  r  at  B  and  DB,  which  is  also  drawn tangent  at  D.  As  you  examine  figure  1-21,  you can deduce that BC is equal to OC. OC is equal to  the  unit  radius,  r. The three functions shown in figure 1-21 are called  the  “direct”   functions.  For  each  direct function  there  is  a  corresponding  “reciprocal” function–meaning  a  function  that  results  when you  divide  1  by  the  direct  function.  You  know that the reciprocal  of any  fraction  is  simply  the fraction   inverted.   Therefore,   for   the   direct function  sine,  which  is  y/r,  the  reciprocal function (called the  cosecant)  is divided by y/r, which  is  r/y. Since  y  at  sine  45°  equals  about  0.7,  the cosecant  for  45°  is  r/y,  which  is  equal  to  1/0.7, or  about  1.4.  The  cosecant  is  indicated  graphically by the line OB in figure 1-21. If you measure this line, you will find that it measures just about 1.4 units  to  the  scale  of  the  drawing. For  the  direct  function  cosine,  which  is  x/r, the reciprocal function (called the secant) is r/x. Since  x  for  cosine  45°  also  measures  about  0.7, it follows that the secant for 45°, r/x, is the same as the cosecant, or also about 1.4. The secant is indicated graphically in figure 1-21 by the line OB also. For the direct function tangent, which is y/x the reciprocal function (called the cotangent)  is x/y.  Since  x  and  y  at  tangent  45°  are  equal,  it follows  that  the  value  for  cotangent  45°  is  the same as that for the tangent, or 1. The cotangent is shown graphically in figure 1-21 by the line DB, drawn  tangent  to  the  circle  at  D. FUNCTIONS  AND  COFUNCTIONS The functions cosine, cosecant, and cotangent are cofunctions of the functions sine, secant, and tangent,  respectively.  A  cofunction  of  an  angle A  has  the  same  value  as  the  corresponding function of (90° – A); that is, the same value as the corresponding function of the complement of the  angle.  The  sine  of  30°,  for  example,  is 0.50000.  The  cosine  of  60°  (the  complement  of 30°)  is  likewise  0.50000.  The  tangent  of  30°  is 0.57735.  The  cotangent  of  60°  (the  complement of 30°)  is  likewise  0.57735. Commonly used functions and cofunctions are as  follows: sin  A  =  cos  (90°  –  A) sec  A  =  csc  (90°  –  A) tan  A  =  cot  (90°  –  A) FUNCTIONS OF OBTUSE  ANGLES In figure 1-22, the point P has generated an obtuse (larger than 90°) angle of 135°. This angle is   the   supplement   of   45°   (two   angles   are supplementary  when  they  total  180°).  We  have left a dotted image of the reference angle A, which is equal to the supplementary angle of 135°. You 1-18