All Roots

Algebra Level pending

$\large \sinh(z)=i$

The roots of the equation can be expressed as $\displaystyle z = i \left(an + \frac{b}{c} \right)d$ , where $n$ belongs to set of integers and $i = \sqrt{-1}$ with $a,b,c,d$ are real constants with $b , c$ as co-primes.

Evaluate $\large \frac{abcd}\pi$ .

The answer is 4.

1 solution

Aman Rajput
Jun 27, 2015

We know that $\sinh(z)=\frac{e^z - e^{-z}}{2}$

Substituting in equation , we get $\frac{e^z - e^{-z}}{2} = \iota$

rearranging , $(e^z - \iota)^2 = 0$ $e^z = \iota$ or $z=\Log(\iota)= 2\iota n\pi + \log(\iota) = 2n\iota\pi + \log(e^{\frac{\iota\pi}{2}})$ or $\boxed{ z = \iota(2n + \frac{1}{2})\pi}$

All Roots

The answer is 4.

1 solution

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