Question

Rewrite the function \(y(t)\) in the form \(y(t)=R e^{a t} \cos (\beta t-\delta)\), where \(0 \leq \delta<2 \pi\). Use this representation to sketch a graph of the given function, on a domain sufficiently large to display its main features. $$ y(t)=\cos \pi t-\sin \pi t $$

Short answer

#Question# Rewrite the function \(y(t) = \cos(\pi t) - \sin(\pi t)\) in the form \(y(t) = Re^{at}\cos(\beta t - \delta)\), with \(0 \leq \delta < 2\pi\), and identify the amplitude, frequency, and phase. Then, sketch the function's graph for the domain [-2π, 2π]. #Answer# The rewritten function is \(y(t) = \sqrt{2}\cos(\pi t - \frac{3\pi}{4})\). The amplitude (R) is \(\sqrt{2}\), the frequency (β) is \(\pi\), and the phase (δ) is \(\frac{3\pi}{4}\). To sketch the graph for the domain [-2π, 2π], start with the regular cosine function, rescale the amplitude, set the frequency to π, and apply a phase shift of \(\frac{3\pi}{4}\) units to the right.

Step-by-step solution

Rewrite the function using trigonometric identities

Recall the trigonometric identity, \(A\cos(\alpha) + B\sin(\alpha) = R\cos(\alpha - \delta)\), where \(R = \sqrt{A^2 + B^2}\) and \(\delta = \tan^{-1}\left(\frac{B}{A}\right)\). Using this identity, rewrite the function as follows: \(y(t) = \cos(\pi t) - \sin(\pi t) = R\cos(\pi t - \delta)\). First, we need to find R and δ: \(R = \sqrt{(\cos{\pi t})^2 + (-\sin{\pi t})^2} = \sqrt{1^2 + (-1)^2} = \sqrt{2}\). And, \(\delta = \tan^{-1}\left(\frac{-\sin{\pi t}}{\cos{\pi t}}\right) = \tan^{-1}(-1) = \frac{3\pi}{4}\). So, we have the rewritten function as: \(y(t) = \sqrt{2}\cos(\pi t - \frac{3\pi}{4})\).

Identify amplitude (R), frequency (a and β), and phase (δ)

From the rewritten function, we can identify the following features: - Amplitude (R): \(\sqrt{2}\), - Frequency (β): \(\pi\) (Since \(y(t) = Re^{at}\cos(\beta t - \delta)\), and there's no \(e^{at}\) term in our function, it means that 'a' is 0, so we focus only on β), - Phase (δ): \(\frac{3\pi}{4}\).

Sketch the function's graph

To sketch the graph, consider the main features identified above (amplitude, frequency, and phase). The amplitude affects the maximum and minimum heights of the graph, frequency determines the length of the cycles, and phase shifts the function horizontally (from left to right for a positive phase value, and from right to left for a negative phase value). To display the main features of the function, we will focus on the domain [-2π, 2π]. For this interval, 1. Start with the regular cosine function. 2. Rescale the amplitude, making the graph achieve a peak at \(\sqrt{2}\) and a trough at \(-\sqrt{2}\). 3. Set the frequency to π, so there's a full cycle in every 2π units. 4. Apply a phase shift of \(\frac{3\pi}{4}\), moving the graph \(\frac{3\pi}{4}\) units to the right. Upon sketching the graph with these features, you will have the graph of \(y(t) = \sqrt{2}\cos(\pi t - \frac{3\pi}{4})\).

Key concepts

Amplitude and Phase Shift

Understanding the amplitude and phase shift is pivotal when graphing trigonometric functions. The amplitude of a trigonometric function is the height from the center line of the graph to its peak or trough. In our exercise, the amplitude is calculated as the square root of the sum of the squares of the coefficients of the cosine and sine functions, which turns out to be \( \sqrt{2} \). A larger amplitude implies a taller wave on the graph.

Phase shift, on the other hand, refers to how much the basic wave is shifted horizontally. This transformation is crucial as it tells us where the waveform starts. It's determined by the angle delta (\( \delta \)) which can be computed using the tangent inverse of the quotient of the sine coefficient over the cosine coefficient. In our example, the phase shift is \( \frac{3\pi}{4} \) units to the right. Graphically, this means that every point on the cosine curve moves \( \frac{3\pi}{4} \) units along the horizontal axis. It's these adjustments that make the depiction of trigonometric functions so rich and varied.

Trigonometric Function Graphing

Graphing a trigonometric function involves adjusting its basic shape based on the amplitude, period, and phase shift. The frequency affects how often the crests and troughs of the wave occur in a given interval; in our case, \( \beta = \pi \), which means one full cycle every \( 2\pi \). With no exponent \( e^{at} \) in our problem, the function isn't experiencing exponential growth or decay, making graphing simpler.

When graphing, it's important to properly scale the axes. Since the amplitude is \( \sqrt{2} \), the vertical axis should be marked accordingly. The frequency tells us that two units on the horizontal scale correspond to a full cycle of the cosine curve. Finally, we account for the phase shift by moving each point on the curve \( \frac{3\pi}{4} \) units to the right. Always begin plotting at the phase shift, setting the wave in motion from there. Plot several cycles to ensure the main features of the function are visible and the graph is representative of the function's behavior over time.

Complex Numbers and Polar Form

Complex numbers are an extension of the real number system, inclusive of solutions to all quadratic equations, even those with no real solutions. They are usually written in the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the square root of -1. The polar form of a complex number is another way to express these numbers, highlighting their magnitude and direction.

The polar form is written as \( r(cos(\theta) + i\sin(\theta)) \), or more compactly, \( re^{i\theta} \), where \( r \) is the magnitude (or modulus), representing the distance from the origin to the point in the complex plane, and \( \theta \) (theta) is the argument (or angle), indicating the direction from the positive real axis to the line connecting the origin with the point. This form makes multiplication and division of complex numbers more intuitive, as you only need to multiply or divide their magnitudes and add or subtract their angles.