Q7P - Mathematical Methods in Physical Sciences

Question

Question: Find each of the following in the x + iy form and compare a computer solution.

$\mathbf{arctan}\mathbf{\left(}\mathit{i}\sqrt{\mathbf{2}}\mathbf{\right)}$

Step-by-Step Solution

Answer

The x + iy form is $z=0.88i+\left(\frac{\pi }{2}\pm n\pi \right)$

Step 1: Given Information

Given expression is, $\arctan \left(i\sqrt{2}\right)$

Step 2: Definition of Trigonometric equation.

A trigonometric equation is one that has one or more trigonometric ratios with unknown angles.

Step 3: Use exponential form to expand the equation

Given the function is, $z=\arctan \left(i\sqrt{2}\right)$.

Write the exponential form of the tan(z).

$$\begin{gathered} \tan \left(z\right)=-i\frac{e^{\left(zi\right)}-e^{\left(-zi\right)}}{e^{\left(zi\right)}+e^{\left(-zi\right)}} \\ -i\frac{e^{\left(zi\right)}-e^{\left(-zi\right)}}{e^{\left(zi\right)}+e^{\left(-zi\right)}}=\sqrt{2}i \end{gathered}$$                 …(1)

 

Simplify equation (1)

 $$\begin{gathered} e^{\left(zi\right)}-e^{\left(-zi\right)}=-\sqrt{2}{e}^{\left(zi\right)}-\sqrt{2}{e}^{\left(-zi\right)} \\ e^{\left(2zi\right)}=\frac{1-\sqrt{2}}{1+\sqrt{2}} \end{gathered}$$   …(2)

 

Take both side in equation (2).

$$\begin{gathered} 2zi=\mathrm{In}\left(\frac{1-\sqrt{2}}{1+\sqrt{2}}\right) \\ =\mathrm{In}(0.17)+i\left(\pi \pm 2n\pi \right) \\ z=-i\frac{\mathrm{In}(0.17)}{2}+\left(\frac{\pi }{2}\pm n\pi \right) \\ z=0.88i+\left(\frac{\pi }{2}\pm n\pi \right) \end{gathered}$$

 

The x + iy form is $z=0.88i+\left(\frac{\pi }{2}\pm n\pi \right)$.