Q24P - Mathematical Methods in Physical Sciences

Question

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

 COS(ix)

Step-by-Step Solution

Answer

The Real part of cos(ix) is cos(x) .

The Imaginary part of cos(ix) is 0.

The absolute value of cos(ix)  is |cos(x) |=cos(x) .

Step 1: Given Information.

The given expression is cos(ix).

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 x of 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(ix).

Let, cos(ix)=cos(iz).

 

Use the formula mentioned below.

sin(iz)=isinh(z)

tan(iz)=itanh(z) 

Divide Left Hand Side of first identity by second identity.

$\frac{\sin \left(\mathrm{iz}\right)}{\tan \left(\mathrm{iz}\right)}=\cos \left(\mathrm{iz}\right)$                                                                                                         …. (1)

Divide Left Hand Side of first identity by second identity.

  $\frac{\mathrm{i} \sinh \left(\mathrm{iz}\right)}{\mathrm{i} \tanh \left(\mathrm{iz}\right)}=\cos h\left(z\right)$                                                                                                     …. (2)

 

Combine equation (1) and equation (2).

cos(iz)=cosh(z)

cos(ix)=cosh(x) 

Hence,

The Real part of cos(ix) is cos(x) .

The Imaginary part of cos(ix) is 0.

The absolute value of cos(ix) is |cos(ix)|=cos(x) .