Q19P - Mathematical Methods in Physical Sciences

Question

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

 $\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{\pi }\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{I}\mathbf{n}\left(2\right)\mathbf{)}$

Step-by-Step Solution

Answer

The value of is $\text{cos(π+i In(2)) is -125}$ .

Step 1: Given Information.

The given hyperbolic number is $\text{cos(π+i In(2)) }$ .

Step 2: Definition of Complex Number.

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

Step 3: Find the value of cos ( 2 i     In   i ) .

Use the identity $\cos \left(x+iy\right)=\cos \left(x\right)\cos h\left(y\right)+i \sin \left(x\right)\sin h\left(y\right)$ .

$$\begin{gathered} \cos \left(\mathrm{\pi }+\mathrm{i} \mathrm{In}\left(2\right)\right)=\cos \left(\mathrm{\pi }\right)\cos h\left(In\left(2\right)\right)+i \sin \left(\mathrm{\pi }\right)\sin h\left(In\left(2\right)\right) \\ =\cos \left(\mathrm{\pi }\right)\cos h\left(In\left(2\right)\right) \\ =-1.25 \end{gathered}$$ 

 

Hence, the value of is $\cos \left(\mathrm{\pi }\right)\cos h\left(In\left(2\right)\right)-1.25$ .