Homework Help

Solve for x: cos (3x) = - cos x

user profile pic

susank7780 | Student, Grade 11 | (Level 1) Salutatorian

Posted May 22, 2013 at 8:17 PM via web

dislike 1 like

Solve for x: cos (3x) = - cos x

2 Answers | Add Yours

user profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted May 22, 2013 at 8:33 PM (Answer #1)

dislike 1 like

Solve cos(3x)=-cosx:

cos(3x)+cosx=0

Use a trigonometric identity to rewrite cos(3x):

`4cos^3(x)-3cos(x)+cos(x)=0`

`4cos^3(x)-2cos(x)=0`

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

(a) ` `2cos(x)=0 ==> cos(x)=0 ==> `x=pi/2+n pi, n in ZZ`

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

`cos^2(x)=1/2`

`cos(x)=+-sqrt(2)/2`

`x=pi/4 + (npi)/2`

---------------------------------------------------------------

The solutions are `x=pi/2 + n pi, pi/4 + (n pi)/2`

---------------------------------------------------------------

The graph of cos(3x)+cos(x) (solutions are zeros):

or the graphs of cos(3x) in black, -cos(x) in red -- the solutions are the intersections:

** If you don't know the identity to get `cos(3x)=4cos^3(x)-3cos(x)` you can use the sum formula:

cos(3x)=cos(2x+x)=cos(2x)cos(x)-sin(2x)sin(x)

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

`=2cos^3(x)-cos(x)-2[(1-cos^2(x))cos(x)]`

`=2cos^3(x)-cos(x)-2cos(x)+2cos^3(x)`

`=4cos^3(x)-3cos(x)`

 

1 Reply | Hide Replies

user profile pic

susank7780 | Student, Grade 11 | (Level 1) Salutatorian

Posted May 23, 2013 at 1:19 AM (Reply #1)

dislike 0 like

What is the identity for cos3x

user profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

Posted May 23, 2013 at 4:56 AM (Answer #2)

dislike 1 like

We have given

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

But

`cos(3x)=4cos^3(x)-3cos(x)` ,so problem reduces to

`4cos^3(x)-3cos(x)=-cos(x)`

`4cos^3(x)-3cos(x)+cos(x)=0`

`4cos^3(x)-2cos(x)=0`

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

`either`

`2cos(x)=0`

`or`

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

`If`

`cos(x)=0`

`` Then   `x=2npi+-pi/2`   , n is an integer.

If  `2cos^2(x)-1=0`

Then

`cos(x)=+-sqrt(1/2)=+-cos(pi/4)`

If

`cos(x)=cos(pi/4)`

`x=2n_1pi+-pi/4 ,`      `n_1` is an integer .

If

`cos(x)=-cos(pi/4)=cos(pi+pi/4)`

`=cos((5pi)/4)`

`x=2n_2pi+-(5pi)/4`  , `n_2` is an integer.

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes