Problem 7 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

\(-\frac{1}{\cosh(1)} + 0i\)

Step 1: Recall the Secant Definition

The secant of a complex number is defined as the reciprocal of the cosine of that number. So, we have: \[\sec(z) = \frac{1}{\cos(z)}\] where \(z = \pi + i\). Therefore, we need to calculate \(\cos(\pi + i)\) first.

Step 2: Use Cosine Addition Formula

For complex numbers, we can use the formula for cosine of a sum: \[\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)\] This means we have to calculate: \( \cos(\pi + i) = \cos(\pi)\cos(i) - \sin(\pi)\sin(i) \).

Step 3: Evaluate Trigonometric and Hyperbolic Functions

First, evaluate the known values: \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \).Next, we need to evaluate the complex arguments: \( \cos(i) = \cosh(1) \) and \( \sin(i) = i\sinh(1) \).

Step 4: Plug in the Values

Substitute these values into the formula:\[\cos(\pi + i) = (-1)\cosh(1) - 0 \sin(i) = -\cosh(1)\]

Step 5: Find Secant

Now that we have \(\cos(\pi + i) = -\cosh(1)\), secant becomes:\[\sec(\pi + i) = \frac{1}{-\cosh(1)} = -\frac{1}{\cosh(1)}\]

Step 6: Final Step: Simplify and Write in Form \(a + ib\)

Since the expression \(-\frac{1}{\cosh(1)}\) is purely real, we can express it as: \[a = -\frac{1}{\cosh(1)},b = 0\]Thus, the form \(a + ib\) is given by:\[-\frac{1}{\cosh(1)} + 0i\]

Key Concepts

Secant Function

The secant function is an important concept in trigonometry. It is defined as the reciprocal of the cosine function. For any complex number \(z\), the secant is expressed as \(\sec(z) = \frac{1}{\cos(z)}\). In simple terms, this means the secant of \(z\) is simply the inverse of its cosine value.
In dealing with complex numbers, the secant function can be a bit more challenging, as it requires us to determine the cosine of complex values first. The secant takes on unique properties when applied to complex numbers, often involving hyperbolic functions and other advanced mathematical components.
In practical scenarios, computing the secant of a complex number like \(\sec(\pi + i)\) requires a series of calculations starting from breaking the complex angle into its separate elements and evaluating each part using known trigonometric identities.

Cosine Addition Formula

The cosine addition formula is a critical tool when working with angles, whether real or complex. The formula is given as \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). This identity allows us to find the cosine of a sum of two angles by breaking it down into simpler trigonometric components.
In the context of complex numbers, the formula becomes even more valuable. It lets us combine the real and imaginary parts of an angle separately. For instance, when used with \(\pi + i\), one evaluates \(\cos(\pi)\) and \(\cos(i)\), as well as \(\sin(\pi)\) and \(\sin(i)\). Each of these terms can be calculated from known trigonometric and hyperbolic values.
This method simplifies complex scenarios by transforming the problem into manageable arithmetic pieces, making it easier to understand and solve problems involving complex arguments.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions for the hyperbola, much like trigonometric functions are for the circle. The two primary hyperbolic functions are the hyperbolic cosine (\(\cosh(x)\)) and hyperbolic sine (\(\sinh(x)\)), defined as:

• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)
• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)

These functions often appear in contexts involving complex numbers because they relate to the exponential function, which defines both \(\cos(i)\) and \(\sin(i)\).
In solving \(\sec(\pi + i)\), hyperbolic functions help to simplify the expressions \(\cos(i) = \cosh(1)\) and \(\sin(i) = i\sinh(1)\). They provide a link between traditional trigonometric identities and the operations needed when dealing with complex arguments. Hyperbolic functions allow us to express complex scenarios in a more straightforward manner, using properties similar to those found in simpler, real-number cases.