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

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embizze's profile pic

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

Posted on

Solve cos(3x)=-cosx:


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




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

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



`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:







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pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

We have given



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











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

If  `2cos^2(x)-1=0`





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




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

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