Question

Show that \(3\left(\frac{\mathrm{e}^{j \omega}-\mathrm{e}^{-j \omega}}{j \omega}\right)=\frac{6 \sin \omega}{\omega}\)

Short answer

Question: Prove that \(3\left(\frac{e^{j \omega}-e^{-j \omega}}{j\omega}\right) = \frac{6 \sin \omega}{\omega}\). Answer: To prove the given expression, we applied Euler's formula to convert the complex exponentials into their equivalent trigonometric functions. After simplification, we found that the expression on the left side is equal to the expression on the right side, thus establishing the equality.

Step-by-step solution

Apply Euler's formula to the complex exponentials

Euler's formula states that \(e^{j \omega} = \cos(\omega) + j\sin(\omega)\) and \(e^{-j \omega} = \cos(\omega) - j\sin(\omega)\). Thus, we can replace complex exponentials with their trigonometric equivalents. So, the expression now looks like: \(3\left(\frac{(\cos(\omega) + j\sin(\omega)) - (\cos(\omega) - j\sin(\omega))}{j\omega}\right)\)

Simplify the expression

Now, we will simplify the expression by carrying out the subtraction and division: \(3\left(\frac{(\cancel{\cos(\omega)} + j\sin(\omega)) - (\cancel{\cos(\omega)} - j\sin(\omega))}{j\omega}\right)\) \(3\left(\frac{2j\sin(\omega)}{j\omega}\right)\)

Cancel out the j terms and simplify further

Now, cancel out the \(j\) term in the numerator and the denominator and simplify: \(3\left(\frac{2\cancel{j}\sin(\omega)}{\cancel{j}\omega}\right)\) \(\frac{6\sin(\omega)}{\omega}\) Now, we see that the expression on the left side has been simplified and is equal to the expression on the right side: \(3\left(\frac{e^{j \omega}-e^{-j \omega}}{j\omega}\right) = \frac{6 \sin \omega}{\omega}\) Thus, the equality is established, and the exercise is solved.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships involving trigonometric functions such as sine, cosine, and tangent. These identities are essential tools in mathematics for simplifying expressions and solving equations.

One of the most popular sets of trigonometric identities involves the Pythagorean identities, which relate the squares of sine and cosine functions:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

Another key identity used in the exercise is the double-angle identity, which specifically relates to the sine function:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

By recognizing these identities, we can transform complex mathematical expressions into simpler forms, as shown with Euler's formula in the solution. Euler's formula itself involves converting complex exponentials into trigonometric terms, showing the strong connection between these concepts.

Knowing these identities allows for simplification of trigonometric expressions, making it easier to solve problems involving angles and frequencies.

Complex Exponentials

Complex exponentials play a crucial role in bridging the gap between trigonometry and calculus. Euler's formula is key here, as it relates complex numbers to trigonometric functions: \

• \( e^{j\omega} = \cos(\omega) + j\sin(\omega) \)
• \( e^{-j\omega} = \cos(\omega) - j\sin(\omega) \)

These expressions reveal how much beauty and symmetry lie within mathematics. Euler's formula helps to convert between exponential and trigonometric expressions, providing a comprehensive way to simplify calculations, especially involving periodic functions.

In the original exercise, complex exponentials are rewritten using their trigonometric equivalences. This conversion is crucial because it allows us to apply the trigonometric identities and simplify the given expression effectively.

Understanding complex exponentials is important not just in mathematics but also in physics and engineering, where they are used to describe oscillations, waveforms, and even quantum mechanics.

Sinusoidal Functions

Sinusoidal functions form the foundation for describing wave-like, periodic phenomena. They are characterized by their smooth, repetitive cycles, typical of sine and cosine functions. These functions are defined as:

• \( y = A \sin(Bx + C) + D \)
• \( y = A \cos(Bx + C) + D \)

The parameters \(A\), \(B\), \(C\), and \(D\) have specific impacts on the graph of the function, affecting amplitude, frequency, phase shift, and vertical shift respectively.

In the exercise, a sinusoidal function appears when converting complex exponentials into their trigonometric form using Euler's formula. Recognizing the sine part of the function enables us to simplify the expression into familiar sinusoidal terms, making further manipulation straightforward.

Sinusoidal functions are not only fundamental in mathematics, but they are also vital in understanding natural phenomena like sound waves, light waves, and alternating current electricity. These functions' ability to model cyclical behaviors makes them an integral concept across various scientific disciplines.