Saturday, September 6, 2008

`cos(2x) - cos(x) = 0` Find the exact solutions of the equation in the interval [0, 2pi).

`cos(2x)-cos(x)=0 , 0<=x<=2pi`


using the identity `cos(2x)=2cos^2(x)-1`


`cos(2x)-cos(x)=0`


`2cos^2(x)-1-cos(x)=0`


Let cos(x)=y,


`2y^2-y-1=0`


solving using the quadratic formula,


`y=(1+-sqrt((-1)^2-4*2(-1)))/(2*2)`


`y=(1+-sqrt(9))/4=(1+-3)/4=1,-1/2`


`:. cos(x)=1, cos(x)=-1/2`


cos(x)=-1/2


General solutions are,


`x=(2pi)/3+2pin, x=(4pi)/3+2pin`


Solutions for the range `0<=x<=2pi` are,


`x=(2pi)/3 , x=(4pi)/3`


cos(x)=1


General solutions are,


`x=0+2pin`


solutions for the range `0<=x<=2pi`  are,


`x=0 , x=2pi`


combine all the solutions ,


`x=0, x=2pi , x=(2pi)/3 , x=(4pi)/3`

No comments:

Post a Comment

What was the device called which Faber had given Montag in order to communicate with him?

In Part Two "The Sieve and the Sand" of the novel Fahrenheit 451, Montag travels to Faber's house trying to find meaning in th...