Results 1 to 1 of 1

Thread: Real and Imaginary parts

  1. #1
    Member kjchauhan's Avatar
    Nov 2009

    Real and Imaginary parts

    Please help me to solve the following :

    Prove that:

    $\displaystyle Sin^{-1}(cosec \alpha) = \left( 2n + (-1)^n \right) \frac{\pi}{2} + i(-1)^n log cot\left(\frac{\alpha}{2} \right)$

    Given that,

    $\displaystyle Sin^{-1}(a+ib)=n\pi + (-1)^n sin^{-1}(a+ib), n\in Z$

    Thanks in advance..

    I got some idea...

    Let $\displaystyle sin^{-1}(cosec\alpha)=x+iy$

    $\displaystyle \therefore cosec\alpha = sin(x+iy)$

    $\displaystyle \therefore cosec\alpha = sin x cosh y + i sinh y cos x$

    on comparing,

    $\displaystyle cosec \alpha =sin x cosh y $ and $\displaystyle 0=sinh y cos x$

    from these, we have,

    $\displaystyle cos x = 0 \Rightarrow x = \frac{\pi}{2}$


    $\displaystyle cosec^2\alpha = (1-cos^2 x)coshy$

    $\displaystyle \Rightarrow cosec\alpha = coshy$

    $\displaystyle \Rightarrow y= cosh^{-1}(cosec\alpha)$

    $\displaystyle \Rightarrow y= log(cosec\alpha+cot\alpha)$

    $\displaystyle \Rightarrow y= log(cot\frac{\alpha}{2})$

    Hence substituting these values of x and y in above we get the result..
    Last edited by kjchauhan; Jan 7th 2011 at 02:08 PM. Reason: Merged posts.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. kindly explain this step of real and imaginary parts
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: Feb 27th 2011, 10:07 AM
  2. Real and Imaginary Parts of a Complex Product
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Oct 5th 2010, 12:13 PM
  3. real and imaginary
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Apr 30th 2010, 03:37 AM
  4. real and imaginary parts of tan(z)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Dec 17th 2009, 05:58 AM
  5. Real / Imaginary Parts
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Feb 28th 2009, 10:30 PM

/mathhelpforum @mathhelpforum