How do you find the general solutions for #2 cos x + 1 = cos x#?

1 Answer
Nov 28, 2015

#x= (2k+1)pi#

Explanation:

Subtract #cos(x)# from both sides:

#2cos(x) +1 -cos(x)=cos(x)-cos(x)#

and summing things up, we have

#cos(x)+1=0#

Now subtract one from both sides:

#cos(x)+1-1=0-1#

and so

#cos(x)=-1#

And since the cosine is the projection of the points of the unit circle on the #x#-axis, the only possible point is the "east pole" of the cirle, which means #x=180^@#, or, in radians, #x=\pi#.

Of course this solution is not unique, since you can do as many additional laps as you like, obtaining all the possible solutions

#x=pi+2kpi = (2k+1)pi#