Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is it possible to calculate the value of $\delta$ from the relation

$\delta=\sin^{-1}(5.4i)$ ? where $i=\sqrt{-1}$

share|improve this question
    
See Inverse Trig Functions –  Amzoti Apr 4 '13 at 15:40
    
...or see this. –  J. M. Apr 4 '13 at 15:40
add comment

2 Answers

Let $\delta=x+iy$

So, $\sin(x+iy)=5.4i$

Now, $\sin(x+iy)=\sin x\cos(iy)+\cos x\sin(iy)=\sin x\cosh y+i\cos x\sinh y$

Equating the real parts, $\sin x\cosh y=0\implies \sin x=0$ as $\cosh y\ge 1$ for real $y$

So, $\cos x=\pm1$

If $\cos x=1, x=2m\pi$ where $m$ is any integer

and $\sinh y=5.4\implies \frac{e^y-e^{-y}}2=\frac{27}5\implies 5(e^y)^2-54e^y-5=0$

Solve for $e^y$ which is $>0$ for real $y$

If $\cos x=-1, x=2(n+1)\pi$ where $n$ is any integer

and $\sinh y=-5.4\implies \frac{e^y-e^{-y}}2=-\frac{27}5\implies 5(e^y)^2+54e^y-5=0$

Solve for $e^y$ which is $>0$ for real $y$

share|improve this answer
add comment

If $\sin(\delta)=5.4i$, then $$\begin{align} \sin(\delta)&=i\sinh(\sinh^{-1}(5.4))\\ &=\sin(i\sinh^{-1}(5.4))\\ \end{align} $$

because it is a fact that $\sin(ix)=i\sinh(x)$. If that is unfamiliar to you, apply the identities $\sin(x)=\frac{\exp(ix)-\exp(-ix)}{2i}$ and $\sinh(x)=\frac{\exp(x)-\exp(-x)}2$.

So $\delta$ would be $$i\sinh^{-1}(5.4)+2\pi n$$ or $$\pi-i\sinh^{-1}(5.4)+2\pi n$$

If you want a firmly defined $\sin^{-1}$, then probably $i\sinh^{-1}(5.4)$ is what you would go with, since that would be consistent with a $\sin^{-1}$ whose range is the part of the complex plane with $-\pi/2\leq\Re(z)\leq\pi/2$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.