Problem 1 - A (terse) introduction to Lebesgue integration

Question

Use Euler's formula which says \(e^{i \theta}=\cos \theta+i \sin \theta\) to derive the trigonometric identities: $$ \begin{aligned} \sin \theta_{1} \cos \theta_{2} &=\frac{1}{2}\left(\sin \left(\theta_{1}+\theta_{2}\right)+\sin \left(\theta_{1}-\theta_{2}\right)\right) \\ \cos \theta_{1} \cos \theta_{2} &=\frac{1}{2}\left(\cos \left(\theta_{1}+\theta_{2}\right)+\cos \left(\theta_{1}-\theta_{2}\right)\right) \\ \sin \theta_{1} \sin \theta_{2} &=\frac{1}{2}\left(\cos \left(\theta_{1}+\theta_{2}\right)-\cos \left(\theta_{1}-\theta_{2}\right)\right) \end{aligned} $$

Step-by-Step Solution

Answer

These identities are derived using Euler's formula by expressing and manipulating complex exponentials.

Step 1: Use Euler's Formula

Start with Euler’s formula: \[e^{i\theta} = \cos \theta + i\sin \theta.\]Express the complex exponentials for two angles \(\theta_1\) and \(\theta_2\) using this formula.

Step 2: Product of Complex Exponentials

Calculate the product:\[ e^{i\theta_1} \cdot e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}.\]Similarly, calculate:\[ e^{i\theta_1} \cdot e^{-i\theta_2} = e^{i(\theta_1 - \theta_2)}.\]

Step 3: Express in Trigonometric Form

Using Euler's formula, expand the product expressions in terms of trigonometric functions:\[ e^{i(\theta_1 + \theta_2)} = \cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2),\]\[ e^{i(\theta_1 - \theta_2)} = \cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2).\]

Step 4: Linear Combinations of Sines and Cosines

To isolate \(\sin(\theta_1) \cos(\theta_2), \cos(\theta_1) \cos(\theta_2),\text{ and }\sin(\theta_1) \sin(\theta_2)\), take the real and imaginary parts of the expressions:The identities:\[\cos \theta_1 \cos \theta_2 = \frac{1}{2} \left( \cos(\theta_1 + \theta_2) + \cos(\theta_1 - \theta_2) \right),\]\[\sin \theta_1 \sin \theta_2 = \frac{1}{2} \left( \cos(\theta_1 - \theta_2) - \cos(\theta_1 + \theta_2) \right),\]\[\sin \theta_1 \cos \theta_2 = \frac{1}{2} \left( \sin(\theta_1 + \theta_2) + \sin(\theta_1 - \theta_2) \right).\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They help in simplifying expressions and solving trigonometric equations. A common set of these identities is the angle sum and difference identities, which allow us to find the sine or cosine of the sum or difference of two angles. Understanding these can seem tricky, but they become friendlier through examples like Euler's formula. Euler's identity, \(e^{i\theta} = \cos \theta + i\sin \theta\), is incredibly useful for deriving various identities, including trigonometric ones. With it, you can derive product-to-sum identities, such as:

• \( \sin \theta_1 \cos \theta_2 = \frac{1}{2} \left( \sin(\theta_1 + \theta_2) + \sin(\theta_1 - \theta_2) \right) \)
• \( \cos \theta_1 \cos \theta_2 = \frac{1}{2} \left( \cos(\theta_1 + \theta_2) + \cos(\theta_1 - \theta_2) \right) \)
• \( \sin \theta_1 \sin \theta_2 = \frac{1}{2} \left( \cos(\theta_1 - \theta_2) - \cos(\theta_1 + \theta_2) \right) \)

These identities are indispensable tools in trigonometry, physics, and engineering, offering simplification in a practical problem-solving environment.

Complex Exponentials

Complex exponentials combine both exponential functions and complex numbers to create powerful mathematical tools. The basics of these are rooted in Euler's formula, \( e^{i \theta} = \cos \theta + i \sin \theta \). This simple yet elegant formula links exponential functions with trigonometric functions.Euler's formula shows how the unit circle can express any angle as a complex exponential. This has profound implications and applications, especially in fields such as electrical engineering and quantum physics. It allows for easy manipulation of trigonometric expressions by converting into exponential forms, and then applying the rules of exponents.Consider multiplying two complex exponentials: \( e^{i\theta_1} \times e^{i\theta_2} = e^{i(\theta_1 + \theta_2)} \). This showcases how the addition of angles works seamlessly in this form. Hence, Euler's approach simplifies otherwise complex trigonometric manipulations and provides a comprehensive understanding of waveforms and oscillations.

Linear Combinations of Sines and Cosines

Linear combinations of sines and cosines involve expressing a function as a sum of these basic trigonometric functions. This is extremely helpful in transforming complex expressions into more usable forms. When you work with Euler's formula, you discover how easily these combinations emerge and simplify calculations.Using identities derived from complex exponentials, one can isolate combinations like:

• \( \sin(\theta_1) \cos(\theta_2) \) and recognize its identity in terms of \( \sin(\theta_1 + \theta_2) \) and \( \sin(\theta_1 - \theta_2) \)
• \( \cos(\theta_1) \cos(\theta_2) \) as a sum of cosines of angles
• \( \sin(\theta_1) \sin(\theta_2) \) as a difference of cosines

By understanding how to break down these expressions, we can apply them to solve differential equations or analyze periodic phenomena efficiently. This conceptual grasp not only aids in simplifying trigonometric problems but enhances analytical skills across numerous mathematical challenges.