Question

Consider the function $$ f(\theta)=\frac{1}{2}(a+i b) e^{i \theta}+\frac{1}{2}(a-i b) e^{-i \theta} $$ where \(a\) and \(b\) are real numbers. Show that \(f\) can be written in the form $$ f(\theta)=C_{1} \cos \theta+C_{2} \sin \theta $$ and determine the values of \(C_{1}\) and \(C_{2}\).

Short answer

#tag_title# Short Answer: Given the complex function \(f(\theta)\), we were able to simplify it and express it in the trigonometric form as \(f(\theta)=C_{1} \cos \theta+C_{2} \sin \theta\), where \(C_1\) is equal to \(a\) and \(C_2\) is equal to \(-b\).

Step-by-step solution

Rewrite the function using Euler's formula

Euler's formula states that \(e^{ix} = \cos x + i \sin x\). By applying this formula, we can rewrite the given function \(f(\theta)\) as $$ f(\theta) = \frac{1}{2}(a+ib)(\cos \theta + i \sin \theta) + \frac{1}{2}(a-ib)(\cos (-\theta) + i \sin (-\theta)). $$ Since \(-\theta\) just results in opposites for sin while the cos is the same, we can go ahead and replace \(-\theta\) with \(\theta\) in our expression to simplify it.

Simplify the expression

Replace the trigonometric functions with their equivalent counterparts and simplify the expression: $$ f(\theta) = \frac{1}{2}(a+ib)(\cos \theta + i \sin \theta) + \frac{1}{2}(a-ib)(\cos \theta - i \sin \theta). $$ Now, distribute the constants to both trigonometric terms in each of the parentheses: $$ f(\theta) = \frac{1}{2}(a \cos \theta + ib \cos \theta + i a \sin \theta - b \sin \theta) + \frac{1}{2}(a \cos \theta - ib \cos \theta - i a \sin \theta - b \sin \theta). $$ Next, combine like terms: $$ f(\theta) = (a \cos \theta - b \sin \theta) + i (b \cos \theta + a \sin \theta). $$ Since a real function does not have imaginary parts, we only need the real part of the function, which gives: $$ f(\theta) = a \cos \theta - b \sin \theta. $$

Express the function in the required form and find the values of \(C_1\) and \(C_2\)

Comparing the simplified function to the required form, we can see that $$ f(\theta) = C_{1} \cos \theta + C_{2} \sin \theta = a \cos \theta - b \sin \theta. $$ Thus, we can conclude that \(C_1 = a\) and \(C_2 = -b\). Therefore, the function can indeed be written in the required form with \(C_1 = a\) and \(C_2 = -b\).

Key concepts

Euler's Formula

Euler's Formula is a fundamental link between trigonometry and complex exponential functions. It expresses complex exponentials in terms of trigonometric functions. This formula is written as \( e^{ix} = \cos x + i \sin x \). Here, \( x \) is a real number, \( \cos x \) is the cosine function, and \( \sin x \) is the sine function, while \( i \) is the imaginary unit.

Euler's formula is particularly useful as it allows us to express combinations of trigonometric functions as complex exponentials, simplifying calculations and solutions in complex analysis. By converting complex numbers into this trigonometric form, we can easily manipulate and simplify expressions. In the exercise above, we used Euler's formula to transform the given function into a form that is easier to interpret and simplify. The ability to switch between exponential and trigonometric forms is crucial in fields such as signal processing and electrical engineering.

With this transformation, we see how the function was reduced to a simpler expression by converting the complex exponentials to a combination of sine and cosine components, leading us to the ultimate goal of finding the coefficients \( C_1 \) and \( C_2 \).

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying expressions involving sine and cosine functions. One important set of identities is related to symmetry properties, such as \( \cos(-x) = \cos x \) and \( \sin(-x) = -\sin x \). These identities demonstrate how the sine function is odd and the cosine function is even, which is foundational in modifying angles in trigonometric expressions.

In our exercise, these symmetries allowed for the simplification of trigonometric parts of the expression when substituting \( -\theta \) with \( \theta \). This ensures that terms are organized logically, enabling the step-by-step simplification. Handling trigonometric identities effectively allows for the reduction of expressions to their simplest forms, making it easier to compare with desired forms for extracting coefficients or other constants.

Having a firm understanding of these identities is vital for working with trigonometric functions in calculus, geometry, and applied mathematics. They provide tools to manipulate and deduce relationships between angles and sides of triangles, which is everywhere in mathematical analysis, physics, and engineering. By leveraging these identities, complex-looking problems can be broken down into manageable parts.

Complex Functions

Complex functions extend the concept of functions to the complex plane, involving variables and values that are complex numbers. These functions can be represented with a real part and an imaginary part, like in the function from the exercise where it is initially given as \( \frac{1}{2}(a+i b) e^{i \theta}+\frac{1}{2}(a-i b) e^{-i \theta} \).

Working with complex functions often involves manipulating these real and imaginary components to achieve a specific goal. In the exercise, we isolated real and imaginary terms to rewrite the original expression solely in terms of real trigonometric functions. This was necessary to express the function \( f(\theta) \) in terms of coefficients \( C_1 \) and \( C_2 \), highlighting how complex numbers are used to simplify equations involving trigonometric functions.

• Real Part: involves purely non-imaginary terms, \( a \cos \theta - b \sin \theta \).
• Imaginary Part: comprises imaginary terms, which can be omitted if focusing on real solutions.

Reasoning through complex functions not only provides powerful insights into the behavior of trigonometric expressions but also reveals pathways for solving equations in various scientific and engineering applications. It allows for a unified approach to solving real-world problems by utilizing the rich structure of complex numbers.