Q31P - Mathematical Methods in Physical Sciences

Question

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

$\mathbf{sinh}\left(\ln 2 +\frac{i \pi }{3}\right)$

Step-by-Step Solution

Answer

The x + iy form of the given equation is $\sinh \left(\ln 2 +\frac{i \mathrm{\pi }}{3}\right)$

Step 1: Given Information.

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

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 x + iy .

Use the formula $\sin h(x+y)=\sin h xh \cos y + \cos h x \sin h y$to break the given question.

 

Consider the following values.

$$\begin{gathered} x= \ln \left(2\right) \\ y=\frac{\mathrm{\pi }}{3} \end{gathered}$$           

$\sin h (x+yi)=\sin h \left(x\right) cps\left(y\right) + i \cos h \left(x\right) \sin \left(y\right)$

Put the value.

$$\begin{gathered} \sin h\left(\ln 2 + \frac{i \mathrm{\pi }}{3}\right)=\sin h \left(\ln 2\right) \cos \left(\frac{\mathrm{\pi }}{3}\right)+i \cos h \left(\ln 2\right) \sin \left(\frac{\mathrm{\pi }}{3}\right) \\ =0.375+1.083 i \end{gathered}$$

 

Therefore, the x + iy form of the given equation is $\sin h\left(\ln 2 + \frac{i \mathrm{\pi }}{3}\right) \mathrm{is} 0.375+1.083 i$