Question

Find each of the following in the x + iyform and check your answers by computer.

$\mathbf{cosh}\left(\frac{\mathrm{i\pi }}{2}-\ln 3\right)$

Short answer

The form of the given equation $\cosh \left(\frac{i\mathrm{\pi }}{2}-\ln 3\right)\mathrm{is}-\frac{4i}{3}$.

Step-by-step solution

Given Information.

The given expression is $\left(\frac{i\mathrm{\pi }}{2}-\ln 3\right)$.

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.

Write in the form of .

Use the formula $\cosh (\mathrm{x}+\mathrm{y})=\cosh \mathrm{x}\cosh \mathrm{y}+\sinh \mathrm{x}\sinh \mathrm{y}$to break the given question.

Consider the following values.

role="math" localid="1658811061300" $$\begin{gathered} x=-\ln 3 \\ y=\frac{\mathrm{\pi }}{2} \\ \cosh (\mathrm{x}+\mathrm{y})=\mathrm{coshx}\cosh \mathrm{y}+\sinh \mathrm{x}\sinh \mathrm{y} \end{gathered}$$

Put the values.

$\cosh \left(\frac{\mathrm{i}\mathrm{\pi }}{2}-\ln 3\right)=\cosh (-\ln 3)\cos \left(\frac{\mathrm{\pi }}{2}\right)+\mathrm{i}\sin \mathrm{h}(-\ln 3)\sinh \left(\frac{\mathrm{\pi }}{2}\right)$

Therefore, the x + iy form of the given equation $\cosh \left(\frac{i\mathrm{\pi }}{2}-\ln 3\right)is\frac{-4i}{3}$.