Problem 3 - Advanced Engineering Mathematics

Question

$$\sin \left(\frac{\pi}{4}+i\right)=\sin \frac{\pi}{4} \cosh (1)+i \cos \frac{\pi}{4} \sinh (1)=1.0911+0.8310 i$$

Step-by-Step Solution

Answer

The solution to \( \sin \left(\frac{\pi}{4}+i\right) \) is approximately \( 1.0911+0.8310i \).

Step 1: Identify the Components

The given expression is \( \sin \left(\frac{\pi}{4}+i\right) \). We need to express it using the formula for the sine of a complex number, which is \( \sin(a + bi) = \sin(a)\cosh(b) + i\cos(a)\sinh(b) \).

Step 2: Apply the Formula

Using the components \( a = \frac{\pi}{4} \) and \( b = 1 \), substitute into the formula: \( \sin \left(\frac{\pi}{4} + i \right) = \sin \frac{\pi}{4} \cosh(1) + i \cos \frac{\pi}{4} \sinh(1) \).

Step 3: Calculate \( \sin \frac{\pi}{4} \) and \( \cos \frac{\pi}{4} \)

We start by calculating \( \sin \frac{\pi}{4} \) and \( \cos \frac{\pi}{4} \). Since both are equal to \( \frac{\sqrt{2}}{2} \approx 0.7071\).

Step 4: Calculate \( \cosh(1) \) and \( \sinh(1) \)

Next, calculate the hyperbolic functions \( \cosh(1) \) and \( \sinh(1) \). The values are \( \cosh(1) \approx 1.5431 \) and \( \sinh(1) \approx 1.1752 \).

Step 5: Substitute Values and Simplify

Now substitute these calculated values back into our expression: \( 0.7071 \times 1.5431 + i \times 0.7071 \times 1.1752 \), which is approximately real part 1.0911 and imaginary part 0.8310.

Step 6: Combine and Conclude

Combine the calculated real and imaginary parts to write the complete solution for the sine of the complex number: \( 1.0911 + 0.8310i \).

Key Concepts

Sine of a Complex Number

Complex numbers are used extensively in advanced mathematics to extend the real number system. In this context, the sine function also extends beyond real numbers to complex numbers. A complex number can be represented as \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). The formula for the sine of a complex number \( z = a + bi \) is:

• \( \sin(a + bi) = \sin(a)\cosh(b) + i\cos(a)\sinh(b) \)

This formula allows us to calculate the sine of any complex number by expressing it in terms of sine and cosine of the real part (\( a \)) and the hyperbolic sine (\( \sinh \)) and cosine (\( \cosh \)) of the imaginary part (\( b \)). This is crucial for problems involving waves and oscillations in complex systems, such as electrical engineering or quantum mechanics.

Hyperbolic Functions

Hyperbolic functions, similar to trigonometric functions, are counterparts for complex numbers. They are defined using exponential functions. The hyperbolic sine and cosine are relevant for computing the sine of a complex number.

• The hyperbolic cosine, \( \cosh(x) \), is defined as \( \frac{e^x + e^{-x}}{2} \).
• The hyperbolic sine, \( \sinh(x) \), is defined as \( \frac{e^x - e^{-x}}{2} \).

These functions appear in many areas of mathematics and physics, particularly when dealing with certain types of differential equations and in the modeling of hyperbolic geometry. They simplify calculations with exponential growth patterns or describe motions such as the path of a hanging cable known as a catenary.

Complex Numbers

Complex numbers are a fundamental part of mathematics, formed by combining a real number with an imaginary number. The notation \( a + bi \) is used, where \( a \) represents the real part and \( bi \) represents the imaginary part.

• The magnitude or modulus of a complex number is given by \( |z| = \sqrt{a^2 + b^2} \).
• The argument of a complex number is the angle \( \theta \) it makes with the positive real axis.

These numbers allow us to solve equations that do not have real solutions, such as \( x^2 = -1 \), whose solution is \( i \). They have essential applications across various fields, including engineering, physics, and computer graphics. Understanding their properties and operations enables complex analysis, leading to insights into waveforms, alternating current analysis, and fractals.

Mathematical Formulas

Mathematical formulas serve as tools that translate complex relationships into manageable expressions for analysis and computation. In complex analysis, formulas are the bridge that connects theoretical math with practical application. The sine formula for complex numbers, for example, \( \sin(a + bi) = \sin(a)\cosh(b) + i\cos(a)\sinh(b) \), simplifies calculations:

• We substitute known values to simplify expressions into results that can be easily computed or graphed.
• They transform abstract mathematics into coherent problem-solving methods, allowing predictions or conclusions about systems or phenomena.

Each formula provides a structured way to express mathematical reasoning, ensuring operations remain consistent and reliable, vital for advancing scientific knowledge and technology.