Q10P - 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.

 $\mathit{a}\mathit{r}\mathit{c}\mathit{c}\mathit{o}\mathit{s}\left(5/4\right)$.

Step-by-Step Solution

Answer

The form x+iy of the given equation arccos(5/4).

$$\begin{gathered} z_1=-i\mathrm{In}\left(2\right)\pm \left(2\mathrm{n\pi }\right) \\ {\mathrm{z}}_2=\mathrm{iIn}\left(2\right)\pm \left(2\mathrm{n\pi }\right) \end{gathered}$$

Step 1: Given Information.

The given expression is $In\left(-e\right)$.

Step 2: Meaning of rectangular form.

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

Step 3: Convert in quadratic equation.

Consider the complex number $z=arc\cos \left(5/4\right)$ .

Rewrite the above expression.

$\cos \left(z\right)=\frac{5}{4}$

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

$\frac{e^{\left(zi\right)}+e^{\left(-zi\right)}}{2}\frac{5}{4}$

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

$$\begin{gathered} u+\frac{1}{u}=\frac{5}{2} \\ {u}^2-2.5u+1=0 \end{gathered}$$

 

Step 4: Solve the quadratic equation.

Write the coefficient and then substitute in the formula.

$$\begin{gathered} a=1 \\ b=-2.5 \\ c=1 \end{gathered}$$

Put in the formula.

$$\begin{gathered} u=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ u=\frac{2.5\pm \sqrt{{\left(2.5\right)}^2-4}}{2} \\ u=\frac{2.5}{2}\pm \frac{1.5}{2} \end{gathered}$$

Step 5: Convert in rectangular form.

Convert in rectangular form.

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

Find the value of $z_1$.

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

Step 6: Convert in rectangular form.

Convert in rectangular form.

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

 

Find the value of $z_2$.

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

 

Hence the general solution of the given equation $arc\cos \left(5/4\right)$.

$$\begin{gathered} z_1=-i \mathrm{In}\left(2\right)\pm \left(2\mathrm{n\pi }\right) \\ {\mathrm{z}}_2=\mathrm{i} \mathrm{In}\left(2\right)\pm \left(2\mathrm{n\pi }\right) \end{gathered}$$