Results 1 to 8 of 8

Math Help - why is cos(-4x)=cos(4x)?

  1. #1
    Banned
    Joined
    Aug 2010
    From
    Singapore
    Posts
    93

    why is cos(-4x)=cos(4x)?

    why is cos(-4x)=cos(4x)? teach me concepts. can add on to sin(-4x) tan(-4x)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by stupidguy View Post
    why is cos(-4x)=cos(4x)? teach me concepts. can add on to sin(-4x) tan(-4x)
    Its because cos(x) is an even function (symmetric about the y axis), meaning that f(-x)=f(x). Now, sin(x) is an odd function (symmetric about the origin), meaning f(-x)=-f(x). You can see why this is the case by looking at the graphs of these two functions.

    Now, tan(x) is an odd function, since \tan (-x)=\dfrac{\sin (-x)}{\cos(-x)}=\dfrac{-\sin x}{\cos x}=-\tan x.

    Does this clarify things?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,751
    Thanks
    652
    Hello, stupidguy!

    \text{Why is }\,\cos(-4x)=\cos(4x)\;?

    We know: . \cos\theta \:=\:\frac{adj}{hyp}


    Look at the graphs:


    Code:
          |
          |           *
          |  hyp   *  |
          |     *     |
          |  * 4x     |
        - * - - - - - * - -
          |    adj


    Code:
          |
          |     adj
        - * - - - - - * - -
          |  * -4x    |
          |     *     |
          |   hyp  *  |
          |           *
          |


    Get it?

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Aug 2010
    From
    Singapore
    Posts
    93
    Quote Originally Posted by Soroban View Post
    Hello, stupidguy!


    We know: . \cos\theta \:=\:\frac{adj}{hyp}


    Look at the graphs:


    Code:
          |
          |           *
          |  hyp   *  |
          |     *     |
          |  * 4x     |
        - * - - - - - * - -
          |    adj


    Code:
          |
          |     adj
        - * - - - - - * - -
          |  * -4x    |
          |     *     |
          |   hyp  *  |
          |           *
          |


    Get it?

    I dun get u!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Unknown008's Avatar
    Joined
    May 2010
    From
    Mauritius
    Posts
    1,260
    Quote Originally Posted by stupidguy View Post
    I dun get u!
    In the first one:
    adjacent is positive, hypotenuse is positive. cos of angle 4x becomes +adj/+hyp = +ve.

    In the second one:
    adjacent is positive, hypotenuse is positive. cos of angle -4x remains +adj/+hyp = +ve.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Aug 2010
    From
    Singapore
    Posts
    93
    Quote Originally Posted by Unknown008 View Post
    In the first one:
    adjacent is positive, hypotenuse is positive. cos of angle 4x becomes +adj/+hyp = +ve.

    In the second one:
    adjacent is positive, hypotenuse is positive. cos of angle -4x remains +adj/+hyp = +ve.
    however u r making an assumption that 4x is smaller than 90 degree. what if it is 135 degree?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2
    One way to see it would be to find the Taylor series expansion of f(x) = \cos{x} about x = 0 and note that all the powers of x are even, but I realise that this is in the pre-Calculus section.
    Quote Originally Posted by Soroban View Post
    Hello, stupidguy!
    And this is why people should sign-up with appropriate names.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    If 4x=135^0, then -4x=360^o-4x=360^o-135^o=225^o

    135^o=180^o-45^o

    225^o=180^o+45^o

    hence, both angles have the same horizontal co-ordinate on the unit-circle.

    Cos(angle)=horizontal co-ordinate of a point on the unit-circle circumference,
    hence it can be located on the horizontal axis.

    If 4x=220^o\Rightarrow\ -4x=360^o-220^o=140^o

    220^o=180^o+40^o

    140^o=180^o-40^o

    so the angles have the same horizontal co-ordinate.


    Think of Cos(angle) as the horizontal co-ordinate of a point on the circle.
    Sin(angle) is the vertical co-ordinate.
    Tan(angle) is the slope of the line going through the origin (centre of unit-circle) and the point on the circle.
    Or Tan(angle) is Sin(angle) divided by Cos(angle).

    A positive angle is an anticlockwise movement starting at zero.
    A negative angle is a clockwise movement starting at 360 degrees.
    Attached Thumbnails Attached Thumbnails why is cos(-4x)=cos(4x)?-cos-4x-.jpg  
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum