Problem 27 - Advanced Engineering Mathematics

Question

$$ \text { Prove that }|\cosh z|^{2}=\cos ^{2} y+\sinh ^{2} x $$

Step-by-Step Solution

Answer

The identity \(|\cosh z|^2 = \cos^2 y + \sinh^2 x\) is proven by expanding and using trigonometric and hyperbolic identities.

Step 1: Understanding Hyperbolic and Trigonometric Identities

First, recall the definitions of the hyperbolic cosine and sine functions:\[ \cosh z = \cosh(x + iy) = \cosh x \cos(y) + i \sinh x \sin(y) \]The modulus squared of a complex number \(a + ib\) is given by \(|a + ib|^2 = a^2 + b^2\). Here, \(a = \cosh x \cos y\) and \(b = \sinh x \sin y\).

Step 2: Write the Modulus Squared of \(\cosh z\)

Calculate the modulus squared of \(\cosh z\):\[ |\cosh z|^2 = (\cosh x \cos y)^2 + (\sinh x \sin y)^2 \]

Step 3: Expand the Expressions

Expand each squared term:\[ (\cosh x \cos y)^2 = \cosh^2 x \cos^2 y \]\[ (\sinh x \sin y)^2 = \sinh^2 x \sin^2 y \]

Step 4: Simplify Using the Identity \(\cosh^2 x - \sinh^2 x = 1\)

Recall the identity for hyperbolic functions: \(\cosh^2 x - \sinh^2 x = 1\).Now substitute into the expression:\[ |\cosh z|^2 = \cosh^2 x \cos^2 y + \sinh^2 x \sin^2 y \]

Step 5: Use Trigonometric Identities

Use the identity \(\cos^2 y + \sin^2 y = 1\) on the terms separately by factoring out:\[ |\cosh z|^2 = \cos^2 y (\cosh^2 x - \sinh^2 x) + \sinh^2 x \]This simplifies to:\[ |\cosh z|^2 = \cos^2 y \cdot 1 + \sinh^2 x \]Thus:\[ |\cosh z|^2 = \cos^2 y + \sinh^2 x \]

Step 6: Conclusion

We see that the final expression \(|\cosh z|^2 = \cos^2 y + \sinh^2 x\) matches what was to be proven. Therefore, the identity holds true for \(|\cosh z|^2\) as required.

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, with each complex number having a real part and an imaginary part. They are often written in the form \(a + ib\), where \(a\) is the real part and \(b\) is the imaginary part. Complex numbers can be visualized in the complex plane, where \(a\) is plotted on the horizontal axis and \(b\) on the vertical axis.
The modulus of a complex number, \(|a + ib|\), represents its distance from the origin in the complex plane and is given by the formula \[ |a + ib| = \sqrt{a^2 + b^2} \].
Squaring the modulus results in \(a^2 + b^2\), which is used in the proof to find the modulus squared of \(\cosh z\). This step is crucial for comparing it with trigonometric and hyperbolic identities.

Trigonometric Identities

Trigonometric identities are key mathematical relationships between trigonometric functions, like sine and cosine.
A fundamental identity to remember is \[ \cos^2 y + \sin^2 y = 1 \], which is often used to simplify expressions and solve equations.
This identity is directly employed in the exercise when proving the equivalency of modulus squared values. By isolating and manipulating the cosine and sine terms, we leverage these identities to simplify the squared expressions in the calculations.
It ultimately helps in structuring the solution to show that \(|\cosh z|^2 = \cos^2 y + \sinh^2 x\), aligning trigonometric and hyperbolic components.

Modulus Squared

Understanding the concept of modulus squared is crucial when dealing with complex numbers.
It's a simplified measure that gives the squared value of the distance from a point to the origin in the complex plane.
For a complex number \(a + ib\), the modulus squared is \(a^2 + b^2\).
In the context of hyperbolic functions, when we analyze \(\cosh z\), where \(z\) is a complex number, we calculate its modulus squared as \[ \cosh^2 x \cos^2 y + \sinh^2 x \sin^2 y \].
This helps in transitioning between hyperbolic identities and arriving at the required solution which shows equivalence between the modulus squared and given identity. The concept allows us to see beyond imaginary components by predominantly focusing on the sum of squares of real-valued parts.

Hyperbolic Identities

Hyperbolic functions, similar to trigonometric functions, have their identities that simplify and relate their behavior. Important hyperbolic identities include \[ \cosh^2 x - \sinh^2 x = 1 \].
This is analogous to the Pythagorean identity in trigonometry. Hyperbolic cosine (\cosh) and hyperbolic sine (\sinh) are the counterparts of regular sine and cosine but defined in terms of exponential functions.
In the exercise, this identity is essential in simplifying and expressing the \(\cosh z\) in terms of known values to prove that \(|\cosh z|^2 = \cos^2 y + \sinh^2 x\).
By leveraging these identities, it becomes straightforward to deduce equivalences and simplify expressions involving hyperbolics and complex numbers, painting a comprehensive picture of how these mathematical functions intertwine.