Problem 5 - Advanced Engineering Mathematics

Question

Express the given quantity in the form \(a+i b\). \(\tan (i)\)

Step-by-Step Solution

Answer

\( \tan(i) = 0 + i \tanh(1) \)

Step 1: Understand the Problem

We need to express \( \tan(i) \) in the form \( a + i b \), where \( a \) and \( b \) are real numbers.

Step 2: Use a Trigonometric Identity

Recall that the tangent of a complex number can be expressed as:\[ \tan(z) = \frac{\sin(z)}{\cos(z)} \text{ for any complex number } z. \]So, we have: \[ \tan(i) = \frac{\sin(i)}{\cos(i)}. \]

Step 3: Calculate \( \sin(i) \) and \( \cos(i) \)

For any complex number \( z = x + i y \),\[ \sin(z) = \sin(x) \cosh(y) + i \cos(x) \sinh(y), \]\[ \cos(z) = \cos(x) \cosh(y) - i \sin(x) \sinh(y). \]Substituting \( z = i \), where \( x = 0 \) and \( y = 1 \), we have:\[ \sin(i) = \sin(0) \cosh(1) + i \cos(0) \sinh(1) = i \sinh(1), \]\[ \cos(i) = \cos(0) \cosh(1) - i \sin(0) \sinh(1) = \cosh(1). \]

Step 4: Simplify the Expression

Now, substitute \( \sin(i) \) and \( \cos(i) \) back into the equation for \( \tan(i) \):\[ \tan(i) = \frac{i \sinh(1)}{\cosh(1)} = i \frac{\sinh(1)}{\cosh(1)} = i \tanh(1). \]

Step 5: Present the Result

The quantity \( \tan(i) \) can be expressed in the form \( a + i b \). We find:- Real part \( a = 0 \)- Imaginary part \( b = \tanh(1) \)Thus, \( \tan(i) = 0 + i \tanh(1) \).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \), is a fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

where \( \theta \) is an angle. For real numbers, the tangent function repeats for every \( \pi \) (180 degrees), indicating it is periodic. The function has vertical asymptotes, meaning it goes to infinity at certain angles because cosine becomes zero. This is crucial in understanding its graph.
When we extend tangent to complex numbers, the formula remains the same. However, we use the properties of hyperbolic sine and cosine since complex numbers can include real and imaginary components. By understanding how tangent behaves with complex numbers, we are able to find expressions involving imaginary components, like \( \tan(i) \).
This forms the basis for calculations in fields like complex analysis, where real and complex numbers interact.

Hyperbolic Functions

Unlike traditional trigonometric functions, hyperbolic functions deal with hyperbolas instead of circles. These functions are analogous to trigonometric functions in many respects:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) (hyperbolic sine)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) (hyperbolic cosine)
• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \) (hyperbolic tangent)

Hyperbolic functions are essential for dealing with complex numbers, especially when such numbers are purely imaginary, like \( i \). They arise when considering the exponential function for complex arguments.
In the solution for \( \tan(i) \), hyperbolic sine and cosine were used to express \( \sin(i) \) and \( \cos(i) \) respectively. The combination resulted in \( \tan(i) \) expressed as \( i \tanh(1) \). This illustrates how hyperbolic functions, despite their scary appearance, simplify trigonometric expressions involving complex numbers.
Understanding hyperbolic functions is key in fields like engineering and physics, where wave patterns and exponential growth are prevalent.

Complex Analysis

Complex analysis is the study of functions that depend on complex numbers. It extends the concepts of calculus to the complex plane, where each point in the plane represents a complex number \( a + ib \) with real part \( a \) and imaginary part \( b \). This field deals not only with complex functions but also with their derivatives and integrals.
In complex analysis, functions such as sine, cosine, and tangent take on a new dimension when applied to complex numbers. An example in the exercise, \( \tan(i) \), needed complex analysis principles for its resolution.

• The use of complex numbers allows for the extension of trigonometric and exponential functions into new territories.
• It enables the evaluation and manipulation of functions across a broader class of inputs.

Complex analysis has applications in various fields, such as electromagnetism, fluid dynamics, and quantum mechanics, where it allows for sophisticated modeling of physical phenomena. The strength of complex analysis lies in its ability to handle difficult problems by simplifying them using the properties of complex numbers.