Question

Evaluate each of the following in$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.${\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{i}}$

Short answer

The value of${(-1)}^{\sin i}$ is $e^{\left[-\sinh \left(1\right)(\pi \pm 2n\pi )\right]}$Where.$n=0,1,2,3,\mathrm{...}$

Step-by-step solution

Given Information. 

The given expression is.${(-1)}^{\sin i}$

Definition of Complex Number and Hyperbolic functions.

The numbers that are presented in the form of$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$where, $\mathit{x}$isthe real numbers and$\mathit{y}$is an imaginary number, those numbers are referred to as called Complex numbers.

The basic hyperbolic functions are:

1. Hyperbolic sine(sinh)
2. Hyperbolic cosine(cosh)
3. Hyperbolic tangent(tanh)

Find the value of.(−1)sini 

It is known that $\sin \left(i\right)=i\sinh \left(1\right)$

$\begin{array}{c}{(-1)}^{i\sinh \left(1\right)}=e^{\left(\ln \left({(-1)}^{i\sin \left(1\right)}\right)\right)}\\ =e^{\left(i\sinh \left(1\right)\ln (-1)\right)}\\ =e^{\left[i\sinh \left(1\right)\left(\ln \left(1\right)+i(\pi \pm 2n\pi )\right)\right]}\\ {(-1)}^{i\sinh \left(1\right)}=e^{\left[-\sinh \left(1\right)(\pi \pm 2n\pi )\right]}\end{array}$

Hence,

The value of ${(-1)}^{\sin i}$is$e^{\left[-\sinh \left(1\right)(\pi \pm 2n\pi )\right]}$ Where$n=0,1,2,3,\mathrm{...}$