+4 Find all solutions for the equation \(sin(x+\frac{\varpi}{6})-sin(x-\frac{\varpi}{6})=\frac{1}{2}\) in the interval \([0, 2\varpi)\).
Thanks so much! : )
+128408 Note that
sin (x + pi/6) = sin(x)cos(pi/6) + sin( pi/6) cos (x)
And
sin ( x - pi/6) = sin (x)cos(pi/6) -sin (pi/6) cos (x)
So we have
sin (x + pi/6) - sin (x - pi/6) = 1/2
[ sin(x)cos(pi/6) + sin( pi/6) cos (x) ] - [ sin(x)cos(pi/6) - sin( pi/6) cos (x) ] = 1/2
sin(pi/6) cos(x) + sin (pi/6) cos(x) = 1/2
(1/2)cos(x) + (1/2) cos (x) = 1/2
cos (x) = 1/2
And this happems at x = pi/3 and x = (5/3) pi on the interval [0,2pi )
