Graphs of sine and cose. 289. Suppose that the unit of length is chosen to represent Let MP, drawn perpendicular to OX, represent the value of sin corresponding to the value OM of 0; then the curve traced out by the point P represents the graph of sin 0. As OM increases from 2 to 4, from 4 to 6π, from 6 to 8π,.... MP passes through the same series of values as when OM increases from 0 to 2π. Since sin (0) - sin 0, the values of MP lying to the left of O are equal in magnitude but are of opposite sign to values of MP lying at an equal distance to the right of 0. Thus the graph of sin is a continuous waving line extending to an infinite distance on each side of 0. The graph of cos is the same as that of sin 6, the origin Graphs of tane and cote. 290. As before, suppose that the unit of length is chosen to represent a radian; then any angle of radians will be represented by a line OM which contains units of length. Let MP, drawn perpendicular to OX, represent the value of tan corresponding to the value OM of 0; then the curve traced out by the point P represents the graph of tan 0. By tracing the changes in the value of tan e as varies from 0 to 2π, from 2 to 4π,......, it will be seen that the graph of tan consists of an infinite number of discontinuous equal branches as represented in the figure below. The part of each branch beneath XX' is convex towards XX', and the part of each branch above XX' is also convex towards XX'; hence at the point where any branch cuts XX' there is what is called a point of inflexion, where the direction of curvature changes. The proof of these statements is however beyond the range of the present work, The various branches touch the dotted lines passing through the points marked The student should draw the graph of cot 0, which is very similar to that of tan 0. Graphs of sece and cosec 0. 291. The graph of sec is represented in the figure below. It consists of an infinite number of equal festoons lying alternately above and below XX', the vertex of each being at the unit of distance from XX. The various festoons touch the dotted lines passing through the points marked The graph of cosec is the same as that of sec 6, the origin П being at the point marked - in the figure. 2 CHAPTER XXIII. SUMMATION OF FINITE SERIES. 292. An expression in which the successive terms are formed by some regular law is called a series. If the series ends at some assigned term it is called a finite series; if the number of terms is unlimited it is called an infinite series. A series may be denoted by an expression of the form where Un+19 = Thus if uncos (a+nẞ), then un+1=cos {a+(n+1)ß} ; and if un=cot 2n-1a, then un+1=cot 2" a. 293. If the 7th term of a series can be expressed as the difference of two quantities one of which is the same function of r that the other is of r+1, the sum of the series may be readily found. For let the series be denoted by U 1 + U2+ Uz + +un ...... and its sum by S, and suppose that any term Up=Vr + 1− Vr ; then S-(2-1)+(V3 − V2)+(V4 −V3)+...+(Vn¬Vn−1)+(Vn + 1 − Vn) =Vn+1-V1• 294. To find the sum of the sines of a series of n angles which are in arithmetical progression. Let the sine-series be denoted by sin a+sin (a+B) + sin (a+28)+......+sin {a+ (n-1) B}. B 2 2n-3 2n-1 2 sin (a+(-1), 5) sin = cos(a+238) - cos (+218) By addition, H. K. E. T. B} β COS COS (a 2.5 sin-cos (a-)-00s (a +213) 2 COS (a− COS |