Results 1 to 2 of 2

Thread: tan identity

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    3

    tan identity

    if anyone could show me how to do this that would be awesome

    show that if: $\displaystyle
    tan(2y)=\frac{1}{tan(x)}\$

    then this is equivalent to the expression:$\displaystyle y=\frac{\Pi}{4}-\frac{x}{2}$

    thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, biker.josh07!

    I have a "visual" solution.



    Show that if: $\displaystyle \tan(2y)\:=\:\frac{1}{\tan x}\$

    then this is equivalent to the expression: $\displaystyle y\:=\:\frac{\pi}{4}-\frac{x}{2}$

    Since $\displaystyle \tan2y \:=\:\frac{1}{\tan x}$, then two angles are complementary.



    Consider this right triangle:


    Code:
                          *
                        * |
                      * x |
                    *     |
                  *       | a
                *         |
              *           |
            * 2y          |
          * - - - - - - - *
                  b

    $\displaystyle \text{Since }\tan 2y \,=\,\frac{a}{b}\:\text{ and }\:\tan x \,=\,\frac{b}{a},\:\text{ then: }\:\tan2y \,=\,\frac{1}{\tan x}$

    . . $\displaystyle x$ and $\displaystyle 2y$ are in the same right triangle.


    Hence: .$\displaystyle x + 2y \:=\:\frac{\pi}{2} \quad\Rightarrow\quad 2y \:=\:\frac{\pi}{2} - x$


    Therefore: .$\displaystyle y \;=\;\frac{\pi}{4} - \frac{x}{2}$

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. identity
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Dec 13th 2009, 02:09 PM
  2. identity 2
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Jul 27th 2009, 11:26 PM
  3. identity
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Jan 21st 2009, 09:40 PM
  4. Sin/Cos Identity
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Sep 1st 2008, 04:41 PM
  5. Identity
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Feb 24th 2008, 11:10 AM

Search Tags


/mathhelpforum @mathhelpforum