Q6P - Mathematical Methods in Physical Sciences

Question

Find each of the following in rectangular form x+iy and check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{c}\mathit{o}\mathit{s}\left(i \mathrm{In}5\right)$.

Step-by-Step Solution

Answer

The rectangular form of the given question  is cos(i In5)$=\frac{13}{5}$.

Step 1: Given Information.

The given expression is $\cos \left(i \mathrm{In}5\right)$.

Step 2: Meaning of rectangular form.

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

Step 3: Convert it into rectangular form.

Use the complex formula $\cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2}$ to rewrite the above expression.

$$\begin{gathered} \cos \left(i \mathrm{In} 5\right)=\frac{e^{i\left(i in 5\right)}+e^{-i\left(i In5\right)}}{2} \\ \cos \left(i \mathrm{In} 5\right)=\frac{e^{i^{2}In5}+e^{-i^2 In5}}{2} \\ \cos \left(i \mathrm{In} 5\right)=\frac{e^{I{n}^{5^{-1}}}+e^{\mathrm{In}5}}{2} \\ \cos \left(i \mathrm{In} 5\right)=\frac{5^{-1}+5}{2} \end{gathered}$$

Step 4: Solve the mathematical expression.

Write the derived expression and solve further.

$$\begin{gathered} \cos \left(i \mathrm{In}5\right)=\frac{\left({\displaystyle \frac{1}{5}}\right)+5}{2} \\ \cos \left(i \mathrm{In}5\right)=\frac{1+25}{2\left(5\right)} \\ \cos \left(i \mathrm{In}5\right)=\frac{26}{2\left(5\right)} \\ \cos \left(i \mathrm{In}5\right)=\frac{\left({\displaystyle \frac{1}{5}}\right)+5}{2} \\ \cos \left(i \mathrm{In}5\right)=\frac{1+25}{2\left(5\right)} \\ \cos \left(i \mathrm{In}5\right)=\frac{13}{5} \end{gathered}$$

Therefore, the rectangular form $\cos \left(i \mathrm{In}5\right)$ of is $\frac{13}{5}$ .