As the terminal line makes a complete revolution, may increase through any one of the intervals: But no matter through which of these intervals @ increases, sin will vary through the same set of values (Theorem, Section 59). Hence the part of the graph in each interval is congruent to that in the interval from 0 to 27, which was plotted above. Starting at the origin, the graph does not begin to repeat until 2π. Hence 2π, or 360°, is the smallest constant such that sin (0 +2π) sin 0, and hence 2, or 360° is the period (Definition, Section 59) of sin 0. The part of the graph from 0 to 2π should be fixed in mind carefully, and the portion corresponding to each quadrant noted, as the graph affords a simple means of remembering the following properties of sin ✪ (see page 42). Zeros of sin 0. or T. For angles less than 27, sin 0 0 if 0 0 = = From the periodicity of the function, the other zeros are therefore Sign of sin 0. Sin 0 is positive if 0 is in the first or second quadrant, negative if is in the third or fourth quadrant. The periodicity shows that this holds whether or not is positive and less than 2π. Maximum and minimum values of sin 0. The maximum value between 0 and 2π is sin (π/2) = 1, and the minimum value sin (3π/2) = 1. The other values of 0 for which sin 0 = 1 may be found by means of the periodicity. Ө Changes of sin 0. As 0 increases from 0 to π/2, sin 0 increases from 0 to 1. 1 to 0. As increases from 3π /2 to 2π, sin 0 increases from Notice that the numerical value of sin 0 cannot exceed unity, and, in particular, that it does not become infinite. Symmetry of the graph. The graph appears to be symmetrical with respect to the origin. Hence, probably, The graphs of cos (Fig. 95) and tan ◊ (Fig. 96) are constructed in like manner, and should be fixed in mind. The various properties of reciprocal functions (page 118) enable us to get certain properties of cot 0, sec 0, and esc 9 directly from the graphs of tan 0, cos 0, and sin respectively. (See also Exercise 2 below.) AA ΥΓ 2 2 FIG. 95. Ө cos 0 - 0.5, 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4 0.9, 0.7 0.7, 0.9, 1 1. Discuss the periodicity, zeros, values of for which the function becomes infinite, sign, maxima and minima, changes and symmetry of cos and tan 0. 2. Sketch on the same axes the graphs of the pairs of functions following, and discuss the second function with respect to the properties listed in Exercise 1. (a) sin and csc 0. (b) cos 0 and sec 0. (c) tan and cot 0. 3. Construct the graphs of the six trigonometric functions on the same axes. 4. What properties of the functions can be inferred from graphs or the same axes of (a) sin 0 and cos 0? (b) tan 0 and cot 0? (c) sec O and csc 0? 5. Describe the motion of a particle on a straight line if its distance s from a fixed point on the line at any time t is given by s sin t. Does such a motion approximate any motion occurring in nature? 6. On the same axes sketch the graphs of the functions: ३ 8. By the addition of ordinates (see Exercise 4, page 44) construct the (e) sin x + 3 cos x. 62. Functions of Complementary Angles. Construct an acute angle and its complement 90° - with their initial 0 lines coinciding with the positive part of the x-axis. On their terminal lines take OP O'P', so that rr'. Then P and P' are symmetrical with respect to OA, the bisector of the first quadrant y (why?), and hence x = y' and y = x'. Y Then P sin (90° - 0) cos 0 (1) The cosine, cotangent and cosecant of an angle are so named because they are respectively the sine, tangent, and secant of the complementary angle, as is shown by these relations. The former functions are called the cofunctions of the latter, respectively, and vice versa. With this terminology, the six relations (1) to (5) may be stated as the Theorem. The functions of any acute angle are equal respectively to the cofunctions of the complementary angle. A method of extending the proofs of these relations for any value of 0, not necessarily acute, will be given in Section 68. 63. Tables of Trigonometric Functions. As a consequence of the theorem in the preceding section, tables of values of the trigonometric functions may be printed in very compact form. Since cos (90° — 0) sin 0, a table of sines of any set of angles is also a table of cosines of the complementary angles. The complements of 0°, 1°, 2°, . . ., 88°, 89°, 90° are respectively 90°, 89°, 88°, 2°, 1°, 0°. Hence: 1. The table of sines on pages 8 and 9 of Huntington's Tables is also a table of cosines if read backward. 2. The sines and cosines of angles from 0° to 45° in the Condensed Table on the inside of the back cover of the Tables are, if read upward, the cosines and sines respectively of the angles from 45° to 90°. In like manner, tan ◊ and cot 0 may be given in one table, and so also may sec 0 and csc 0. Thus the theorem on functions of complementary angles makes it practicable to reduce by one-half the space devoted to a table of trigonometric functions. Fractional parts of a degree, in Huntington's Tables, are given in tenths and hundredths instead of in minutes and seconds. One of the merits of this decimal method of subdivision is that the process of interpolation, in finding a function of 0, is identical with that used earlier in the other tables (page 121). B 0.4321 sin E F A FIG. 98. 0.4332 I H D a function of O is given, is illustrated in the examples following. |