Let f be the function defined by fx sin 2x sin x Find the

Let f be the function defined by f(x) = sin 2x - sin x. Find the zeros of the function in the interval {x : -2pi x 2pi}.

Solution

f(x) = sin(2x) - sinx

zero\'s ==> f(x) = 0

==> sin(2x) - sinx = 0

==> 2sinxcosx - sinx = 0 ; since sin(2x) = 2sinxcosx

==> sinx(2cosx - 1) = 0

==> sinx = 0 or 2cosx - 1 = 0

sinx = 0 ==> x = -2 , - , 0 , , 2

2cosx - 1 = 0 ==> cosx = 1/2

==> x = -5/3 , -/3 , /3 , 5/3

Hence solutions are x = -2 , -5/3 , - , -/3 , 0 , /3 , , 5/3 , 2