Q8P - Mathematical Methods in Physical Sciences

Question

Find each of the following in the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form and compare a computer solution.

$\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\left(5/3\right)$.

Step-by-Step Solution

Answer

The $x+iy$ form of the given equation  $arc\sin \left(5/3\right)$ is $\pm i\mathrm{In}\left(3\right)+\left(\frac{\mathrm{\pi }}{2}\pm 2\mathrm{n\pi }\right)$

Step 1: Given Information.

The given expression is $arc\sin \left(5/3\right)$ .

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x+iy in which x is the real part and y is the imaginary part.

Step 3: Convert in quadratic equation.

Consider the complex number. $z=arc\sin \left(5/3\right)$

Rewrite the above expression.

$\sin \left(z\right)=\frac{5}{3}$

 

Write the formula for  $\sin \left(\theta \right)$.

$\frac{e^{\left(zi\right)}-e^{\left(-zi\right)}}{2i}=\frac{5}{3}$

Put$e^{\left(zi\right)}=u$

$$\begin{gathered} u-\frac{1}{u}=\frac{10i}{3} \\ {u}^2-\frac{10i}{3}u-1=0 \end{gathered}$$

 

Step 4: Solve the quadratic equation.

Write the coefficient and then substitute in the formula.

$$\begin{gathered} a=1 \\ b=-\frac{10i}{3} \\ c=1 \end{gathered}$$

Put in the formula.

$$\begin{gathered} u=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \frac{10i}{3}\pm \sqrt{-\frac{100}{9}+4} \\ u=\frac{{\displaystyle \frac{10i}{3}}\pm {\displaystyle \frac{8i}{3}}}{2} \\ u=\frac{5i}{3}\pm \frac{4i}{3} \end{gathered}$$

Step 5: Convert in rectangular form.

Convert in rectangular form.

$$\begin{gathered} zi=In\left(u_1\right) \\ zi=In\left(r\right)+i\left(\theta +2n\pi \right) \\ zi=In\left(\frac{5i}{3}+\frac{4i}{3}\right) \\ zi=In\left(3\right)+i\left(\frac{\pi }{2}\pm 2n\pi \right) \end{gathered}$$ 

Find the value of  $z_1$.

$$\begin{gathered} z_1=\frac{zi}{i} \\ {z}_1=\frac{In\left(3\right)+i\left({\displaystyle \frac{\pi }{2}}\pm 2n\pi \right)}{i} \\ {z}_1=-i \mathrm{I}n\left(3\right)+\left(\frac{\pi }{2}\pm 2n\pi \right) \end{gathered}$$

Step 6: Convert in rectangular form.

Convert in rectangular form.

$$\begin{gathered} zi=In\left(u_2\right) \\ zi=In\left(r\right)+i\left(\theta +2n\pi \right) \\ zi=In\left(\frac{i}{3}\right) \\ zi=In\left(\frac{1}{3}\right)+i\left(\frac{\pi }{2}\pm 2n\pi \right) \end{gathered}$$

Find the value of $z_2$ .

$$\begin{gathered} z_2=\frac{zi}{i} \\ {z}_2=\frac{-In\left(3\right)+i\left({\displaystyle \frac{\pi }{2}}\pm 2n\pi \right)}{i} \\ {z}_2=i In\left(3\right)+\left(\frac{\pi }{2}\pm 2n\pi \right) \end{gathered}$$

Put both the solution in one.

 $z_g=\pm i In\left(3\right)+\left(\frac{\pi }{2}\pm 2n\pi \right)$

 

Hence the general solution is $z_g=\pm i In\left(3\right)+\left(\frac{\pi }{2}\pm 2n\pi \right)$.