Q36P - Mathematical Methods in Physical Sciences

Question

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

$\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(1+\frac{i\pi }{2}\right)$

Step-by-Step Solution

Answer

The x + iy form of the given equation $sinh(1+\frac{\mathrm{i\pi }}{2})$ is 1.543 i .

Step 1: Given Information.

The given expression is $sinh(1+\frac{\mathrm{i\pi }}{2})$.

Step 2: 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.

Step 3: Write in the form of .

Use the formula sinh(x + y) = sinh xh cos y +cosh x sinh y to break the given question.

 

Consider the following values.

                  $$\begin{gathered} x=1 \\ y=\frac{\pi }{2} \end{gathered}$$

sinh(x + yi) = sinh (x) cos(y) + i cosh(x) sinh(y) 

Put the value.

$$\begin{gathered} \sin h\left(1+\frac{i\pi }{2}\right)=\sin h\left(1\right)\cos \left(\frac{\pi }{2}\right)+i \cos h\left(1\right)\sin \left(\frac{\pi }{2}\right) \\ =1.543 i \end{gathered}$$

Therefore, the x + iy form of the given equation $\sin h\left(1+\frac{i\pi }{2}\right)$ is 1.543 i.