Q4MP - Mathematical Methods in Physical Sciences

Question

Find one or more values of each of the following complex expressions and compare with a computer solution.

$\mathbf{(}\mathbf{1}\mathbf{+}\frac{\mathbf{\pi i}}{\mathbf{2}}\mathbf{)}$

Step-by-Step Solution

Answer

The value of $\sin h\left(1+\frac{\mathrm{\pi i}}{2}\right)$ is, 1.54i .

Step 1: Given Information.

The given hyperbolic number is $\left(1+\frac{\mathrm{\pi i}}{2}\right)$ .

Step 2: Definition of Complex Number.

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

Step 3: Find the value of s i n h ( 1 + π i 2 ) .

Use the following mentioned identities for solving the question.

$$\begin{gathered} \sin h\left(yi\right)=i\sin \left(y\right) \\ \cos h\left(yi\right)=\cos \left(y\right) \\ \sin h(x+yi)=\sin h\left(x\right)\cos h\left(yi\right)+\cos h\left(x\right)\sin h\left(yi\right) \\ =\sin h\left(x\right)\cos \left(y\right)+i\cos h\left(x\right)\sin \left(y\right) ......\left(1\right) \end{gathered}$$

 

Substitute,

$$\begin{gathered} x=1 \\ y=\frac{\mathrm{\pi }}{2} \end{gathered}$$

 in equation (1).

$$\begin{gathered} \sin h\left(1+\frac{\mathrm{\pi }}{2}i\right)=\sin h\cos \left(\frac{\mathrm{\pi }}{2}\right)+i\cos h\left(1\right)\sin \left(\frac{\mathrm{\pi }}{2}\right) \\ =i \cos h\left(1\right) \\ =1.54i \end{gathered}$$ 

Hence, the value of $\sin h\left(1+\frac{\mathrm{\pi }}{2}i\right)$ is $1.54i$ .