400 



PRACTICAL MATHEMATICS 



Curve (1) shows y l = 2 sin x 



(2) shows t/ 2 = 2 sin x sin 2x 



2 . 



(3) shows y.j = 2 sin # sin 2x + - sin Sx 



o 



2 1 



(4) shows y => 2 sin a; sin 20 +- sm 3# - sin 4# 



O w 



212 



(5) shows r/5 = 2 sin x sin 2<r + - sin 3x - sin 4tx + - sin 5x 



9 m JO 



and it should be noticed how the addition or subtraction of a 

 sine function brings the resultant curve nearer to the straight line 



EXAMPLES XXII 



(1) If y = a sin qt and x = b sin (qt c) where t is time and 



271 



a, q, b, c are constants ; if q = -^- where T is the periodic time. 



Find the average value of xy during the time T. (B. of E., 1908.) 



(2) Express sin at cos bt as the sum of two terms and integrate 



with regard to t. If a is -= and b is Sa, what is the value of the 



integral between the limits and T ? (B. of E., 1913.) 



(3) If y = a sin qt and x = b cos (qt c) where t is time and 

 a, q, b, c are constants. Find the average value of xy during the 



