Q8MP - Mathematical Methods in Physical Sciences

Question

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

 $\mathit{c}\mathit{o}\mathit{s}\left[2iIn\frac{1-i}{1+i}\right]$

Step-by-Step Solution

Answer

The value of  $\cos \left[2i In\frac{1-i}{1+i}\right] is-1$.

Step 1: Given Information

The given expression is $\cos \left[2i In\frac{1-i}{1+i}\right]$ .

Step 2: Definition of Complex Numbers

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

Step 3: Find the value of z 1 and z 2 .

Consider $w=\cos \left[2i In\frac{1-i}{1+i}\right]$.                                                                    …. (1)

Use cos$\left(zi\right)$=cosh(z) to rewrite the equation (1).

$w=\cos h\left(2In\frac{1-i}{1+i}\right)$

 Let be, $1-i$ be, $z_1$.

$z_1=1-i$

The value of modulus of $z_1$is, $\sqrt{1^2+1^2}=\sqrt{2}$ .

The argument of is $\left(\frac{-1}{1}\right)=-\frac{\mathrm{\pi }}{4}$.

 Let $1-i$be $z_1$.

$z_1=1-i$

The value of modulus of $z_1$is , $\sqrt{1^2+1^2}=\sqrt{2}$ .

The argument of $z_1$ is arctan$\left(\frac{-1}{1}\right)=-\frac{\mathrm{\pi }}{4}$.

Let 1 + i be $z_2$

$z_2=1+i$

The value of modulus of $z_2$ is $\sqrt{1^2+1^2}=\sqrt{2}$ .

The argument of $z_2$ is arctan $\left(\frac{1}{1}\right)=\frac{\mathrm{\pi }}{4}$ .

Step 4: Use the value of z 1 and z 2 to find the desired result.

Put the values of $z_1$and $z_2$ in equation (1).

$$\begin{gathered} w=\cos h\left(2In\frac{\sqrt{2}{e}^{-\mathrm{\pi i}/4}}{\sqrt{2}{e}^{\mathrm{\pi i}/4}}\right) \\ =\cos h\left(2In\left(e^{-\mathrm{\pi i}/2}\right)\right) \\ =\cos h\left[2i\left(-\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)\right] \\ =\cos \left(-\mathrm{\pi }\pm 4\mathrm{n\pi }\right) \\ W=\cos \left(-\mathrm{\pi }\right) \\ W=-1 \end{gathered}$$

Hence the value of $\cos \left[2iIn\frac{1-i}{1+i}\right] is -1$