Q22P - Mathematical Methods in Physical Sciences

Question

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

$\mathbf{sin}\mathbf{(}\mathbf{i}\mathbf{ }\mathbf{ln}\left(\frac{\sqrt{3}+i}{2}\right)\mathbf{)}$

Step-by-Step Solution

Answer

The value of  $\sin \left[\mathrm{i} \ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right]$is -0.5 .

Step 1: Given Information.

The given number is $\sin \left[\mathrm{i} \ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right].$

Step 2: Definition of Complex Number.

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

Step 3: Find the value of sin [ i   ln ( 3 + i 2 ) ] .

Let, $u=\sin \left[\mathrm{i} \ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right]$

Use the identity  $\sin \left(iz\right)=i \sin h\left(z\right)$ .

$$\begin{gathered} u=\sin \left[\mathrm{i} \ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right] \\ =i \sin h\left[ \ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right] \\ =i \sin h\left[In\left(r\right)+i\left(\theta \pm 2n\pi \right)\right] \end{gathered}$$

 

Find the argument and the angle of $\frac{\sqrt{3}+i}{2}$ .

$$\begin{gathered} r=\left|\frac{\sqrt{3}+i}{2}\right| \\ =1 \\ \theta =\frac{\pi }{6} \end{gathered}$$

 $$\begin{gathered} \mathrm{u}=\mathrm{i} \sinh \left[\ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right] \\ =\mathrm{i} \sinh \left[\ln \left(1\right)+\mathrm{i}(\frac{\mathrm{\pi }}{6}\pm 2\mathrm{n\pi })\right] \\ =\mathrm{i} \mathrm{sinhi}(\frac{\mathrm{\pi }}{6}\pm 2\mathrm{n\pi }) \\ =-\sin (\frac{\mathrm{\pi }}{6}\pm 2\mathrm{n\pi }) \\ =-\frac{1}{2} \\ \mathrm{n}=0,1,2,3,.... \end{gathered}$$

Hence, the value of $\sin \left[\mathrm{i} \ln \left(\frac{\sqrt{3}+\mathrm{i}}{2}\right)\right]$ is -0.5