Problem 14 - Advanced Engineering Mathematics

Question

\(u\left(e^{i \theta}\right)=\cos 2 \theta\)

Step-by-Step Solution

Answer

The function maps \(e^{i \theta}\) to \(\cos 2 \theta\) using trigonometric identities.

Step 1: Understanding the Function

We're given the complex function \(u(e^{i \theta}) = \cos 2 \theta\). This function takes an exponential form \(e^{i \theta}\) and outputs the cosine of twice that angle, \(\theta\).

Step 2: Analyzing the Input

The input to the function is \(e^{i \theta}\), which is a point on the unit circle in the complex plane. This effectively means that \(\theta\) is the angle formed with the positive real axis, and \(e^{i \theta} = \cos \theta + i \sin \theta\).

Step 3: Applying the Function

The function takes \(e^{i \theta}\) and transforms \(\theta\) to \(2 \theta\), doubling the angle before applying the cosine function on it. The output will be \(\cos 2 \theta\).

Step 4: Using Trigonometric Identity

Recall the double-angle identity for cosine: \(\cos 2\theta = 2\cos^2 \theta - 1\) or \(\cos 2\theta = 1 - 2\sin^2 \theta\). These identities help us understand that \(\cos 2 \theta\) is directly calculated using \(\cos \theta\) and \(\sin \theta\).

Step 5: Interpretation of Result

The function \(u(e^{i \theta}) = \cos 2\theta\) can be interpreted as a mapping from each point on the unit circle to a specific value based on the angle \(\theta\). This gives us a real number output (between -1 and 1) that represents the cosine of twice the input angle.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and hold true for every value of the variable within their definition domain. They are helpful tools in simplifying expressions and solving equations. For example, the double-angle identity for cosine, \( \cos 2\theta = 2\cos^2 \theta - 1 \), can be used to express \( \cos 2\theta \) in terms of \( \cos \theta \). This is valuable in complex analysis, especially when evaluating trigonometric functions at various angles.

• Basic identities: These include fundamental identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal identities: For example, \( \sin \theta = \frac{1}{\csc \theta} \).
• Pythagorean identities: Derived from the Pythagorean theorem.

Understanding these identities allows us to manipulate and simplify expressions that involve trigonometric functions, which is essential in various fields like wave functions and electrical engineering.

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a useful concept in trigonometry and complex analysis as it helps visualize the relationships between angles and trigonometric values. Each point on the unit circle can be represented as \( e^{i\theta} = \cos \theta + i \sin \theta \), where \( \theta \) is the angle from the positive x-axis.

• Points are defined as \((\cos \theta, \sin \theta)\).
• The angle \(\theta\) is measured in radians, usually between \(0\) and \(2\pi\).
• The unit circle allows easy derivation of trigonometric values for key angles.

The unit circle not only aids in understanding angles but also provides insight into periodic behaviors and symmetry in trigonometric functions, crucial for modeling oscillations and waves.

Double-Angle Formulas

The double-angle formulas are special cases of trigonometric identities used to express functions of \(2\theta\) in terms of \(\theta\). These formulas are particularly useful when dealing with problems involving periodic functions, making calculations more straightforward.

• Cosine double-angle formula: \( \cos 2\theta = 2\cos^2 \theta - 1 \) or \( \cos 2\theta = 1 - 2\sin^2 \theta \).
• Sine double-angle formula: \( \sin 2\theta = 2\sin \theta \cos \theta \).
• Tangent double-angle formula: \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \).

By using these formulas, complex trigonometric expressions can be significantly simplified. They are particularly prevalent in calculus and analysis when evaluating integrals or series involving trigonometric functions.