Problem 11 - Advanced Engineering Mathematics

Question

$$\sinh \left(1+\frac{\pi}{3} i\right)=\sinh (1) \cos \frac{\pi}{3}+i \cosh (1) \sin \frac{\pi}{3}=0.5876+1.3363 i$$

Step-by-Step Solution

Answer

\(\sinh(1 + \frac{\pi}{3} i) = 0.5876 + 1.3363 i\) is the final result.

Step 1: Start with the Formula

Recall that \( \sinh(z) \) for a complex number \( z = x + yi \) is defined as \( \sinh(z) = \sinh(x) \cos(y) + i \cosh(x) \sin(y) \). We apply this formula to \( z = 1 + \frac{\pi}{3} i \).

Step 2: Substitute into the Formula

Substitute \( x = 1 \) and \( y = \frac{\pi}{3} \) into the \( \sinh \) formula:\[\sinh(1 + \frac{\pi}{3} i) = \sinh(1) \cos(\frac{\pi}{3}) + i \cosh(1) \sin(\frac{\pi}{3})\]

Step 3: Calculate \( \sinh(1) \) and \( \cosh(1) \)

Using known values:- \( \sinh(1) \approx 1.1752 \) - \( \cosh(1) \approx 1.5431 \)

Step 4: Calculate Trigonometric Values

Calculate the trigonometric functions:- \( \cos(\frac{\pi}{3}) = \frac{1}{2} \)- \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \approx 0.8660 \)

Step 5: Combine Results

Substitute these values back into the formula:\[\sinh(1 + \frac{\pi}{3}i) = 1.1752 \cdot \frac{1}{2} + i \cdot 1.5431 \cdot 0.8660\]

Step 6: Simplify the Expression

Perform the calculations:- Real part: \( 1.1752 \times 0.5 \approx 0.5876 \)- Imaginary part: \( 1.5431 \times 0.8660 \approx 1.3363 \)Thus,\[\sinh(1 + \frac{\pi}{3}i) = 0.5876 + 1.3363i\]

Step 7: Conclusion: Final Result

The value of \( \sinh(1 + \frac{\pi}{3} i) \) has been calculated and matches the given result \( 0.5876 + 1.3363 i \).

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers formed by combining a real part and an imaginary part. A complex number is generally expressed in the form of \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
This allows complex numbers to represent quantities in two dimensions, helping describe phenomena that can't be captured by real numbers alone.
In many mathematical applications, including hyperbolic functions and trigonometry, complex numbers are crucial. With them, you can solve equations that have no real solutions, as well as work with wave functions in physics and engineering. For instance, the formula for the hyperbolic sine of a complex number:

• \( z = x + yi \)
• \( \sinh(z) = \sinh(x)\cos(y) + i\cosh(x)\sin(y) \)

provides a way to extend hyperbolic functions into the complex plane, facilitating analysis in more advanced fields.

Trigonometric Functions

Trigonometric functions are foundational in mathematics and are used to describe relationships between the angles and sides of triangles. They include sine, cosine, and tangent, and can extend to complex numbers.
The trigonometric values such as \( \cos(\frac{\pi}{3}) \) and \( \sin(\frac{\pi}{3}) \), calculated in the step by step solution, are derived from the unit circle, a tool that maps out the values of these functions at various angles measured in radians.
These functions are not limited to angles in geometry; their applications spread to modeling periodic phenomena like sound waves, light waves, and tides. When dealing with complex numbers, the trigonometric functions extend using Euler's formula, allowing us to relate exponential functions to trigonometric functions seamlessly. This connection is vital for transforming real-world waveforms and signals into a usable mathematical format.

Euler's Formula

Euler's Formula is a bridge between exponential functions, complex numbers, and trigonometry. It states that for any real number \( x \),\[ e^{ix} = \cos(x) + i\sin(x) \]This simple yet profound equation is remarkable because it brings together exponential growth, oscillation, and the imaginary unit into a unified framework.
Euler's formula is frequently used in engineering, physics, and applied mathematics to solve problems involving wave equations, electrical circuits, and signal processing. By expressing trigonometric functions in terms of exponentials, computations become more manageable, especially when dealing with derivatives and integrals of oscillatory functions.
In more advanced mathematics, Euler's formula is fundamental in the study of Fourier transforms, which decompose functions into their constituent frequencies, offering insights into their structure and characteristics.