okay this problem stumped me for a while but here is my work for it, and I just got stuck at the end so if any help can be provided thanks in advance. sin3x = sinx sin(2x + x) = sinx sin2x cos x + cos2x sinx = sinx 2sinx cosx cosx + (2cos^2(x) -1) sinx = sinx 2sinx cos^2(x) + 2cos^2(x) sinx - sinx = sinx 2sinx cos^2(x) + 2cos^2(x) sinx - 2sinx = 0 2sinx (cos^2(x) + cos^2(x) - 2) = 0 2sinx (2cos^2(x) - 2) = 0 4sinx(cos^2(x) -1) = 0 4sinx(-sin^2(x)) = 0 4sinx = 0 , -sin^2x = 0 x = 0 , x = 0, pi, 2pi Why is it when I graph the equation sin3x - sinx = 0 on my calculator, it comes with 7 solutions when i only have 3?
solution sin(2x+x)-sinx=0 sin2xcosx + cos2xsinx-sinx = 0 2sinxcos^2 x + (cos^2 x - sin^2 x) sinx - sinx = 0 2sinxcos^2 x + cos^2xsinx - sin^3 x -sinx=0 sinx(2cos^2 x + cos^2 x - sin^2 x - 1) = 0 sinx(3cos^2x - sin^2x -1) =0 so u kno sinx= 0, thus 0, 180, and 360 degrees are three of the answers then for inside parenthesis 3cos^2x - sin^2x - 1 = 0 3cos^2x - (1- cos^2x) - 1 = 0 3cos^2x - 1 + cos^2x - 1 = 0 4cos^2x - 2 = 0 cos^2x = 1/2 so u kno Cos^-1( plus/minus sq root of 2 / 2) = 45, 315, 35, 225 degrees :zzz: loong problems, well sorta
There is a much simpler method to solve problems like that. If sin(x) = sin(y) then either y=x=2k*pi or y=(pi-x)+2k*pi, where k is integer (zero included). y=3x now, so either 3x=x+2k*pi --> x = k*pi or 3x=(2k+1)*pi -x -->x=(2k+1)*pi/4 ehild
Maybe you already know this, but just in case, sin[m * pi] = 0 for all m. So, x=m*pi for all m satisfies the (trivial) equation: sin[m*pi] = sin[3*m*pi] = 0 -- edit: where m is an integer.
The problem is that the use of your double angle was incorrect but the idea of solving the proble is correct the thing to do here is sin3x=sinx implies sin(2x + x)=sinx implies sin2xcosx +cos2xsinx - sinx=0 implies 2sinx.cosx.cosx +cos2xsinx - sinx=0 implies sinx(2cos^2(x)+ cos2x - 1)=0 implies sinx(2cos^2(x) + 2cos^2(x) - 1 -1)=0 implies sinx(4cos^2(x) - 2)=0 now using the zero product law implies sinx=0 or cos^2(x)=1/2 then the equation will solve to be x=0 + n360 or x=+or- 45 + n360 where n lies in Z or integers from there you will sub in integers the will give you solutions that lie in your domain you draw your graph