Question

Express \(\cos \omega t\) in terms of exponential functions.

Short answer

Question: Express the cosine function \(\cos \omega t\) in terms of exponential functions using Euler's formula. Answer: \(\cos \omega t = \frac{1}{2}(e^{i\omega t} + e^{-i\omega t})\)

Step-by-step solution

Recall Euler's Formula

Euler's formula states that \(e^{ix} = \cos x + i \sin x\). This formula connects complex exponentials with trigonometric functions.

Write the complex conjugate of Euler's formula

The complex conjugate of Euler's formula is obtained by replacing \(i\) with \(-i\) in the formula: \(e^{-ix} = \cos x - i \sin x\)

Add the two equations

Now, add the Euler's formula and its complex conjugate together: \(e^{ix} + e^{-ix} = (\cos x + i \sin x) + (\cos x - i \sin x)\) This simplifies to: \(e^{ix} + e^{-ix} = 2\cos x\)

Express the cosine function in terms of exponential functions

Finally, solve for \(\cos x\) by dividing both sides by 2: \(\cos x = \frac{1}{2}(e^{ix} + e^{-ix})\)

Apply the result to \(\cos \omega t\)

Now, we can replace \(x\) with \(\omega t\) to express \(\cos \omega t\) in terms of exponential functions: \(\cos \omega t = \frac{1}{2}(e^{i\omega t} + e^{-i\omega t})\) So, \(\cos \omega t\) can be expressed as \(\frac{1}{2}(e^{i\omega t} + e^{-i\omega t})\) in terms of exponential functions.

Key concepts

Exponential Functions

Exponential functions are mathematical functions where a constant base is raised to the power of a variable. They are powerful tools in both pure and applied mathematics because they model growth and decay processes, like population growth or radioactive decay.
The general form of an exponential function is written as \( a^x \), where \( a \) is the base and \( x \) is the exponent. The base is usually a positive constant different from 1. One important special case is the natural exponential function \( e^x \), where \( e \) is Euler's number, approximately 2.718.
Euler's number \( e \) is used extensively in calculus as it has unique properties. For instance, the derivative of \( e^x \) is \( e^x \), and its integral is also \( e^x \). These properties make calculations involving \( e \) smooth and predictable.
In the context of complex analysis, Euler's Formula links exponential functions with trigonometric functions through complex numbers. This relationship demonstrates how the exponential function can describe oscillations and waves, fundamental to understanding phenomena like sound and light.

Complex Numbers

Complex numbers extend the concept of numbers to include 'imaginary' numbers. These numbers are mathematically expressed in the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
The utility of complex numbers comes from their ability to solve equations that do not have real solutions. A great example is the square root of a negative number, like \( \sqrt{-1} \), which is defined to be \( i \).
Complex numbers are fundamental in electrical engineering and physics. They simplify calculations involving oscillations and alternating currents.
Euler's formula itself is a fascinating link between complex numbers and other areas of math. It shows that \( e^{ix} = \cos x + i \sin x \). With this, complex numbers provide a more complete understanding and representation of trigonometric concepts.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable involved. They are crucial for simplifying expressions and solving equations involving angles.
Some of the basic trigonometric identities include the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), which describes the relationship between the sine and cosine of the same angle. Similarly, the angle sum and difference identities can be used to expand trigonometric expressions in terms of different angles.
Using Euler's formula, we can express trigonometric functions like cosine and sine as combinations of complex exponential functions. As shown in the step-by-step solution for expressing \( \cos \omega t \) in terms of exponential functions, we use:

• \( e^{i x} = \cos x + i \sin x \)
• \( e^{-i x} = \cos x - i \sin x \)

Adding these equations results in \( e^{ix} + e^{-ix} = 2\cos x \). This identity provides a powerful tool for converting trigonometric expressions into exponential forms, allowing for easier manipulation in calculus and differential equations.