A very convenient unit angle 55. Graphs in Radian Measure. for many such graphs is the radian (§ 43). We shall agree to use the radian as a unit in trigonometric graphs, unless something is said to the contrary. The graph of y sin x, drawn on this scale, is shown in Fig. 56; it resembles Fig. 55 very closely, since 1 radian = 57°.3 is very close to 60°, which was used in § 54. The use of the radian as the unit angle both in such graphs and in all other connections, is universal in the Calculus and in other advanced mathematical topics. Instead of computing 56. Mechanical Construction for sin x. values of y for certain values of x as in the preceding table and plotting these for points on the curve y = sin x, we can shorten the work materially by the following graphical method. Construct a pair of rectangular axes and choose a scale unit; for the sake of fixing our ideas let us suppose that this unit is one inch. At C, a convenient point on the x-axis as center, construct a unit circle. Choose some number for x, and lay off ACB = x radians (above the x-axis if x > 0, below if x < 0). On the x-axis lay off OD arc AB = x in. (to the right if x > 0, to the left if x < 0). Then the abscissa of D, and therefore of every point on the vertical line through D, is x; it is obvious from the construction that the ordinate of B, and therefore of every point on the horizontal line through B, is sin x. Therefore the coördinates of the point P where these lines intersect are (x, sin x), and P is a point on the curve y sin x. This method may be used to plot the curve y = sin x as fol = = lows. Suppose it is desired to plot the curve from x=-π/2 to x/2. Choose a scale unit and lay off on the x-axis OE =1 unit, OH = (7/3) units = 22/21 of OE, approximately. Divide OH into a convenient number of parts (say 4), and then mark the points/2, -5 π/12, - π/3, π/4, π/6, -π/12, π/12, π/6, π/4, π/3, 5 π/12, π/2, on the x-axis. At C, a convenient point on the x-axis, construct a circle with radius =OE. With a protractor (or by bisection of arcs in this case) lay off arcs AB1, AB2, etc. which subtend angles π/12 radians, π/6 radians, etc. at the center C; also, AB', AB, etc., subtending angles/12 radians, /6 radians, etc. Intersections of corresponding horizontal and vertical lines give points. on the curve. Proceed similarly for any value of x. EXERCISES XXIII. GRAPHS OF THE TRIGONOMETRIC FUNCTIONS 1. From the values of cos x in Exs. 11 and 12, p. 78, plot the graph of the equation y = cos x, choosing the units as in § 54. 2. Draw the curve y = cos x by a mechanical construction similar to that of § 56, with the scale used in §§ 55-56. 3. Plot the graphs of each of the equations on the scale of § 54, or on the scale of § 55. [NOTE. The scale of § 55 (radian measure) is preferable. See Table V.] 4. Trace the variations of cos x as x increases from 0 to 2π radians, from the curve of Ex. 2. 5. Trace the variation of tan x as x varies from 0 to 2, from the curve of Ex. 4 (α). Trace the variation of ctn x. 6. Plot the graphs, preferably on the radian scale, of the equations : (a) y = secx, (b) y = csc x. 7. Show that the graph of y = sin x + cos x can be constructed by mechanically adding the ordinates of the two curves y sin x and y = cos x. = 8. By analogy to Ex. 7 show how to draw mechanically each of the 9. Show that the graph of y = x + sin x can be constructed mechani 10. Show how to construct the graph of y shortening the horizontal lengths in the graph of y sin 2x mechanically by = sin x in the ratio 1 : 2. 11. By analogy to Exs. 7, 9, 10, draw mechanically each of the following curves from graphs previously drawn: 12. Show that the graph of y = sec x can be drawn mechanically from that of y = cos x by means of the relation sec x = 1/cos x. 13. By analogy to Ex. 12 show how to draw, from the graphs of sin x and cos x, the graphs of each of the following curves: (a) y = csc x. (b) y = tan x. (c) y = ctn x. (d) y =vers x. 15. Show that the graph of y = sin(x — π/6) can be drawn by moving the graph of y = sin x to the right by an amount π/6. 16. By analogy to Ex. 15 draw mechanically the following curves : 17. Show how to draw mechanically the graphs of each of the following curves: = tan2x. (a) y = sin2x. (b) y = cos2x. (c) y 18. If a point M moves in a circular path of unit radius with a constant angular speed 1 radian per second, show that the angle = t (radians); hence show that the coördinates (x, y) of M are: x = cos t, y = sin t. 2 sin (4 x — π/3). Y M A PART III. APPLICATIONS OF LARGE ANGLES 57. Composition and Resolution of Forces. As in § 48, the components on the axes of any force of magnitude F which makes an angle a with the posi axes. = we may find the components of each of them on each of the The sum of the two x-components is F' + F"z F' cos a'+F" cos a", and is equal to the x-component of the resultant R of F'' and F", as is evident from a figure, since. Proj, OC = ON = OM + MN = Proj, F' + Proj, F". Hence the x component R, of R is: These results hold, by (1), when F' and F" lie in any positions. From (2) and (3) the magnitude R of the resultant and the angle which it makes with the positive x-axis are given by (4) 2 R = √R2 + R2, tan 0 R, R2; = where, in case of ambiguity, the quadrant in which lies is determined by the signs of R, and R, in an obvious manner. 58. The Projection Theorem. We can now generalize the preceding results and prove the following important theorem: The sum of the projections on any straight line 1, of a broken line whose segments are taken in order so that the terminal point of each segment is the initial point of the next, is equal to the projection on l of a line segment joining the initial point of the first segment of the broken line to the terminal point of the last segment. Proof. (1) The theorem is true when the broken line consists of two segments, for using the notation of § 14, in any = figure, Proj, AB = A'B', Proj, BC= B'C', Proj, AC A'C' and A'B' + B'C' = A'C', whatever the order of points A'B'C'. (2) If the theorem is true for n − 1 segments it is also true for n segments. Let A4, be the straight line joining the initial point of the first segment with the terminal point of the (n-1)th segment. Then by hypothesis: ... Proj, A12+ ProjĮ AA, + + Proji An-1 A„ = Projɩ A1A„· Now by (1), Proj, 414, + Proj, 4,4+1 Proj, A14+1; hence, = n+l Proj, A12+... Proj, An-14, + Proj, An4n+1 = Proj; A14+1 A2 A1 A2 FIG. 61 An-1 AA “n+l (3) The theorem is true for 2 segments; hence it is true for 3 segments, consequently for 4 segAn+1 ments, etc., for any number. An 59. Application to Forces and Velocities. The results of § 58 apply to forces and velocities, since the resultant of any number of forces (or velocities) is found by forming a broken line whose sides are equal and parallel to the given forces (or velocities), the initial point of each force (or velocity) being placed at the terminal point of the preceding one. The resultant is represented by the directed line segment connecting the initial point of the first to the terminal point of the last. The result of § 58, applied to forces, exactly as in § 57, gives the components of the resultant R on the two axes, in terms of given forces F, F, F, etc., which make angles a', a", a""', etc., with the positive end of the x-axis, as follows: The magnitudė R of the resultant and the angle ◊ which it makes with the positive x-axis are given by (3) R=√R22+ R2, tan 0 = R, R, The signs of R, and R, determine the quadrant in which lies. |