The above equation is of type #color(red)(asinx+bcosx=c)# where #a, b, and c# are constants. To solve the above equation, we always #color(red)(Divide)# it by #color(red)sqrt(a^2+b^2).# Then the equation will be reduced to the form #sin(A+-B) or cos(A+-B)#.
Here, #2sinx-cosx=3/sqrt(2)# where #a=2 and b=-1#
Now, #sqrt(a^2+b^2)=sqrt(2^2+(-1)^2##=sqrt(5)#
Dividing the equation by #sqrt(5)# we get
#rarrsinx*2/sqrt(5)-cosx*1/sqrt(5)=(3/sqrt(2))/sqrt(5)=3/sqrt(10)#
or, #sinx*cos26.57°-cosx*sin26.57°=sin71.57°#
or, #sin(x-26.57°)=sin71.57°#
or, #x-26.57°=360°*n+(-1)^n*71.57°#where #ntoZZ#
or, #x=360°*n+(-1)^n*71.57°+26.57°# where #ntoZZ#
So, the general solution of the eqaution is
#x=360°*n+(-1)^n*71.57°+26.57°# where #ntoZZ#
