we need to sum the roots of the equation sin2x=92β between x=0 and x=2Ο. These roots, all of which must be roots of the given equation, are
x=2arcsin92ββ,2Οβarcsin92ββ,22Ο+arcsin92ββ, and 23Οβarcsin92ββ
and their sum is 3Ο.
OR
For any b>2 the solutions of y2βby+1=0 are y1β,y2β=2bΒ±b2β4ββ, which are distinct and positive. These solutions are reciprocals because their product must be 1. Since the solutions yiβ=tanxiβ are reciprocals,
Thus, y1β and y2β are tangents of two distinct, complementary, first-quadrant angles, x1β and x2β=2Οββx1β. Since tan(x+Ο)=tanx, there are four values of x between 0 and 2Ο:x1β,Ο+x1β,2Οββx1β and 23Οββx1β. Their sum is 3Ο.