Q27P - Mathematical Methods in Physical Sciences

Question

Question: Find the real part, the imaginary part, and the absolute value of 

sinh(4 + 3i) .

Step-by-Step Solution

Answer

The Real part of sinh(4 + 3i) is - 7.72 .

The Imaginary part of sinh(4 + 3i)   is - 6.55.

The absolute value of sinh(4 + 3i)  is 10.05.

Step 1: Given Information.

The given expression is sinh(4 + 3i).

Step 2: Definition of Complex Number and 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: Find the real , imaginary , absolute value of sinh(4 + 3i) .

Let u = sinh(4 + 3i) .

 

Use cosine formula.

sinh(x + yi) = sinh(x)cosh(y) + icos(x)sinh(y)    …….(1)

 

Use x = 4 and y = 3 in the formula .

 $$\begin{gathered} \mathrm{Re}\left(\mathrm{u}\right)=\sin \left(\mathrm{x}\right)\cos h\left(\mathrm{y}\right) \\ =\sin \left(4\right)\cos h\left(3\right) \\ =-7.62 \\ \mathrm{Im}\left(\mathrm{u}\right)=\cos \left(\mathrm{x}\right)\sin h\left(\mathrm{y}\right) \\ =\cos \left(4\right)\sin h\left(3\right) \\ =-6.55 \\ \left|\mathrm{u}\right|=\sqrt{{\mathrm{Re}}^2\left(\mathrm{u}\right)+{\mathrm{Im}}^2\left(\mathrm{u}\right)} \\ =\sqrt{{\left(-7.62\right)}^2{\left(-6.55\right)}^2} \\ =10.05 \end{gathered}$$

Hence,

The Real part of sinh(4 + 3i) is - 7.62.

The Imaginary part of sinh(4 + 3i) is - 6.55

The absolute value of sinh(4 + 3i) is 10.05.