Q26P - Mathematical Methods in Physical Sciences

Question

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

cosh(2 - 3i) .

Step-by-Step Solution

Answer

The Real part of cosh(2 - 3i) is - 3.72 .

The Imaginary part of cosh(2 - 3i)  is - 0.51.

The absolute value of cosh(2 - 3i)  is 3.755.

Step 1: Given Information.

The given expression is cosh(2 - 3i).

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: Find the real , imaginary , absolute value of cosh(2 - 3i) .

Let u = cosh(2 - 3i) .

 

Use cosine formula.

cosh(x + yi) = cosh(x)cos(y) + isinh(x)sin(y)    …….(1)

 

Use x = 2 and y = -3 in the formula .

 $$\begin{gathered} \mathrm{Re}\left(\mathrm{u}\right)=\cosh \left(\mathrm{x}\right)\cos \left(\mathrm{y}\right) \\ =\cosh \left(2\right)\cos \left(-3\right) \\ =-3.72 \\ \mathrm{Im}\left(\mathrm{u}\right)=\sinh \left(\mathrm{x}\right)\sin \left(\mathrm{y}\right) \\ =\sinh \left(2\right)\sin \left(-3\right) \\ =-0.51 \\ \left|\mathrm{u}\right|=\sqrt{{\mathrm{Re}}^2\left(\mathrm{u}\right)+{\mathrm{Im}}^2\left(\mathrm{u}\right)} \\ =\sqrt{{\left(-3.72\right)}^2{\left(-0.51\right)}^2} \\ =3.755 \end{gathered}$$

Hence,

The Real part of cosh(2 - 3i) is - 3.72.

The Imaginary part of cosh(2 - 3i)  is - 0.51.

The absolute value of cosh(2 - 3i)  is 3.755.