Q18P - Mathematical Methods in Physical Sciences

Question

Evaluate each of the following in x + iyform, and compare with a computer solution.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{2}\mathit{i}\mathbf{ }\mathit{I}\mathit{n}\mathbf{ }\mathit{i}\mathbf{)}$

Step-by-Step Solution

Answer

The value of is $\cos \left(2\mathrm{iIni}\right)\mathrm{is}-1$.

Step 1: Given Information.

The given number is $\cos \left(2iIni\right)$.

Step 2: Definition of Complex Number.

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

Step 3: Find the value of c o s ( 2 i   In   i ) .

Use the identity cos$\left(iz\right)$=$\cosh \left(\mathrm{z}\right)$ .

$\cos \left(2\mathrm{i} \mathrm{In} \mathrm{i}\right)=\cosh \left(2\mathrm{In} \mathrm{i}\right)$

                    $$\begin{gathered} =\cos h\left[2\left(In 1+i\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)\right)\right] \\ =\cos h\left[2i\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)\right] \\ =\cos \left(2i In i\right)=-1 \end{gathered}$$ 

Hence, the value of is $=\cos \left(2i In i\right)=-1$.