Find the general solution of sinx+cosx=1

Question

I think to use the auxillary method though I am unsure on what to do when the equation becomes sin(x+pie/4)=???

Answer 1

Put y=sinx, then cosx=sqrt(1-y^2), so y+sqrt(1-y^2)=1. sqrt(1-y^2)=1-y, 1-y^2=1-2y+y^2, 2y^2-2y=0 so y(y-1)=0, and y=0 or 1. Therefore sinx=0 or 1 and x=(4n+1)(pi)/2 or 2n(pi) ((4n+1)90 or 360n degrees).

This solution is the simple fact that when sinx=1 (when x=90 degrees) cosx=0; similarly when x=0 sinx=0 and cosx=1. The periodic nature of sin and cos means an infinite number of solutions.