Question

Solve `cos^3x=cosx`?

`Cos^3x=cosx` I get got the `0, pi/2, pi` but I cant get `(3pi)/2`. If anyone can help out I would really appreciate it.

Answer 1

See below

Explanation:

It's a little tricky, but try setting the equation = 0

`cos^3x=cosx`

`cos^3x-cosx=0`

Now factor out `cosx`

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

And now factor the difference of two squares

`cosx(cosx+1)(cosx-1)=0`

Set each factor equal to 0 and solve using your unit circle

1) `cosx=0, x={pi/2, (3pi)/2}`

2) `cosx+1=0`

`cosx=-1, x=pi`

3) `cosx-1=0`
`cosx=1, x=0`

Answer 2

See below.

Explanation:

`cos^3(x)=cos(x)`

`cos^3(x)-cos(x)=0`

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

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

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

Pythagorean identity:

`cos(x)(sin^2(x))=0`

`:.`

`cos(x)=0=>pi/2,(3pi)/2`

`sin^2(x)=0=>0, pi,2pi`