Q5MP - Mathematical Methods in Physical Sciences

Question

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

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathbf{\left(}\frac{\mathbf{\pi i}}{\mathbf{4}}\mathbf{\right)}$

Step-by-Step Solution

Answer

The value of  $\tanh \left(\frac{\mathrm{\pi i}}{4}\right)$is, i .

Step 1: Given Information.

The given hyperbolic number is $\tanh \left(\frac{\mathrm{\pi i}}{4}\right)$.

Step 2: Definition of Complex Number.

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

Step 3: Find the value of t a n h ( πi 4 ) .

Use the following mentioned identities to solve the question.

$$\begin{gathered} \sin h\left(yi\right)=i \sin \left(y\right) \\ \cos h\left(yi\right)=\cos \left(y\right) \\ \tan h\left(\frac{\mathrm{\pi i}}{4}\right)=\frac{\sin h\left({\displaystyle \frac{\mathrm{\pi i}}{4}}\right)}{\cos h\left(\frac{\mathrm{\pi i}}{4}\right)} \\ =\frac{\sin h\left({\displaystyle \frac{\mathrm{\pi i}}{4}}\right)}{\cos h\left(\frac{\mathrm{\pi i}}{4}\right)} \\ =i \end{gathered}$$

Hence, the value of $\tan h\left(\frac{\mathrm{\pi i}}{4}\right) is i.$