Results 1 to 5 of 5

Math Help - tan(a*x)=?

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    14

    tan(a*x)=?

    hello
    does anyone knows that is there any sequence for tan(a*x) ?
    for example for tan(2*x)=2*tan(x)/(1-tan^2(x))
    and tan 3x = 3tanx - (tan x)^3]/[1 - 3(tanx)^2]
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Apr 2011
    Posts
    3
    here you are: (follwing is a link)

    prouver que tan(nx) = ... (forum ilemaths)

    Bernard
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,715
    Thanks
    634
    Hello, abdolah!

    Does anyone knows that is there any sequence for tan(nx) ?

    for example: .tan(2x) .= .2*tan(x) / [1 - (tan x)^2]

    and: tan 3x .= .[3*tanx - (tan x)^3] / [1 - 3(tan x)^2]

    Back in college, I discovered a delightful pattern for this.



    Example: tan(5x).


    Consider the expansion of (1 + tan x)^5:

    . . 1 + 5(tan x) + 10(tan x)^2 + 10(tan x)^3 + 5(tan x)^4 + (tan x)^5


    Write alternate terms in the denominator and numerator of a fraction.
    . . Start with the denominator.

    . . . . 5(tan x) . . . . 10(tan x)^3 . . . . (tan x)^5
    . .--------------------------------------------------------
    . . 1 . . . . 10(tan x)^2 . . . . 5(tan x)^4


    Insert alternating signs in the numerator and in the denominator:

    . . . .+5(tan x) . . . . - 10(tan x)^3 . . . . + (tan x)^5
    . .--------------------------------------------------------
    . .+1 . . . . - 10(tan x)^2 . . . . +5(tan x)^4


    . . . . . . . . . . . . . . . . .5(tan x) - 10(tan x)^3 + (tan x)^5
    Therefore: .tan(5x) .= .-----------------------------------------
    . . . . . . . . . . . . . . . . . . 1 - 10(tan x)^2 + 5(tan x)^4

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,715
    Thanks
    634

    This technique can be used for \sin(nx) and \cos(nx)


    Example: . \sin(5x) and \cos(5x)


    Consider the expansion of: .(cos x + sin x)^5

    . . (cos x)^5 + 5(cos x)^4(sin x) + 10(cos x)^3(sin x)^2

    . . . . . . + 10(cos x)^2(sin x)^3 + 5(cos x)(sin x)^4 + (sin x)^5


    Write alternate terms in two rows; start with the second row.

    . . . . . . 5(cos x)^4(sin x) . . . . 10(cos x)^2(sin x)^3 . . . . (sin x)^5

    . . (cos x)^5 . . . . 10(cos x)^3(sin x)^2. . . . 5(cos x)(sin x)^4


    Insert alternating signs in the first row and in ths second row:

    . . . . . .+5(cos x)^4(sin x) . . . .-10(cos x)^2(sin x)^3 . . . .+(sin x)^5

    . .+(cos x)^5 . . . .-10(cos x)^3(sin x)^2. . . .+5(cos x)(sin x)^4


    And we have:

    . .sin(5x) .= .5(cos x)^4(sin x) - 10(cos x)^2(sin x)^3 + (sin x)^5

    . .cos(5x) .= .(cos x)^5 - 10(cos x)^3(sin x)^2 + 5(cos x)(sin x)^4

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    14
    thanks , very good answers
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum