Problem 12 - Advanced Engineering Mathematics

Question

$$\cosh (2+3 i)=\cosh (2) \cos (3)+i \sinh (2) \sin (3)=-3.7245+0.5118 i$$

Step-by-Step Solution

Answer

The solution is \( -3.7245 + 0.5118i \). The calculated result matches the given one.

Step 1: Understanding Complex Cosine Hyperbolic Function

The problem gives the function \( \cosh(z) \) where \( z = 2 + 3i \). \( \cosh(z) \) is defined as \( \cosh(a + bi) = \cosh(a) \cos(b) + i \sinh(a) \sin(b) \). Our task is to compute this expression step by step.

Step 2: Break Down the Given Function Parts

We first note that \( a = 2 \) and \( b = 3 \). We need to calculate the values of \( \cosh(2) \), \( \cos(3) \), \( \sinh(2) \), and \( \sin(3) \).

Step 3: Calculate Hyperbolic and Trigonometric Components

Using a calculator:- \( \cosh(2) \approx 3.7622 \)- \( \cos(3) \approx -0.9899 \)- \( \sinh(2) \approx 3.6269 \)- \( \sin(3) \approx 0.1411 \)

Step 4: Substitute Values into Formula

Insert the calculated values into the equation:\[ \cosh(2+3i) = (3.7622)(-0.9899) + i(3.6269)(0.1411) \]

Step 5: Calculate the Real Part

Multiply the real components:\( 3.7622 \times -0.9899 \approx -3.7245 \).

Step 6: Calculate the Imaginary Part

Multiply the imaginary components:\( 3.6269 \times 0.1411 \approx 0.5118 \).

Step 7: Write Final Answer with Real and Imaginary Parts

Combine the two parts to get the final result: \( \cosh(2+3i) = -3.7245 + 0.5118i \). This confirms the given solution.

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical constructs that allow us to solve equations that have no real solutions. They are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \).

• The \( a \) component is known as the real part.
• The \( bi \) component is known as the imaginary part.

Complex numbers can be visualized as points in a plane called the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
When dealing with functions of complex numbers, such as the hyperbolic cosine function in the exercise above, both the real and imaginary parts contribute to the final result.

Trigonometric Functions

Trigonometric functions are essential mathematical tools used to relate the angles of a triangle to the lengths of its sides. These functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others.

• They are periodic functions, meaning they repeat values over specific intervals.
• Sine and cosine functions have ranges from -1 to 1, making them very useful in modeling wave-like phenomena.

In the context of the exercise with \( \cosh(2+3i) \), the cosine and sine functions are used to decompose the expression into parts that are purely real and purely imaginary.
This allows for calculating the contributions of each component separately, making the manipulations with complex numbers more manageable.

Cosine Hyperbolic Function

The cosine hyperbolic function, denoted \( \cosh(x) \), is one of the hyperbolic functions, which are analogs of the ordinary trigonometric functions but for a hyperbola instead of a circle.

• The formula for \( \cosh(x) \) is \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
• It shares some properties with the regular cosine function, such as being an even function (i.e., \( \cosh(-x) = \cosh(x) \)).

In the original exercise, when we compute \( \cosh(2+3i) \), the real part \( \cosh(2) \) and the imaginary transformations involving cosine and sine reveal the dynamic nature of this function.
It highlights how hyperbolic functions and complex numbers can combine to create intricate and interesting results.