Q30P - Mathematical Methods in Physical Sciences

Question

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

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{3\pi i}{4}\right)$

Step-by-Step Solution

Answer

 $\left(\tan h\frac{3\pi i}{4}\right)$Can be written in the form of $0-i=i$ .

Step 1: Given Information.

The given expression is $\tan h\left(\frac{3\pi i}{4}\right)$ .

Step 2: Definition of Complex Numberand Hyperbolic functions.

The domain of a function refers to the range of values that can be plugged into it. This is the set of x values in a function like f (x). The range of a function is the set of values that the function can take. This is the set of values that the function produces when we enter x   value. The y values are what you're looking for.

The basic hyperbolic functions are:

1. Hyperbolic sine(sinh)
2. Hyperbolic cosine(cosh)
3. Hyperbolic tangent(tanh)

 

Step 3: Write t a n h ( 3 x i 4 ) in the form of x + i y .

Let $u=\tan h\left(\frac{3\pi i}{4}\right)$

It is known that $\tan h\left(zi\right)=i\left(\tan \left(z\right)\right)$ and

$\tan \left(\frac{3\pi }{4}\right)=-1$

$$\begin{gathered} u=i\tan \left(\frac{3\pi }{4}\right) \\ =i\left(-1\right) \\ =-i \end{gathered}$$

$\tan \left(\frac{3\pi }{4}\right)=-1$ 

Hence,$\tan h\left(\frac{3\pi i}{4}\right)$   can be written in the form of $0-i=-i$.