Problem 68 - Precalculus Mathematics for Calculus

Question

Adding an Echo A digital delay device echoes an input signal by repeating it a fixed length of time after it is received. If such a device receives the pure note \(f_{1}(t)=5 \sin t\) and echoes the pure note \(f_{2}(t)=5 \cos t,\) then the combined sound is \(f(t)=f_{1}(t)+f_{2}(t)\) (a) Graph \(y=f(t)\) and observe that the graph has the form of a sine curve \(y=k \sin (t+\phi)\) (b) Find \(k\) and \(\phi\)

Step-by-Step Solution

Answer

The values are \(k = 5\sqrt{2}\) and \(\phi = \frac{\pi}{4}\).

Step 1: Understand the Problem

The task is to combine the functions \(f_1(t) = 5 \sin t\) and \(f_2(t) = 5 \cos t\) and express the combined signal \(f(t)\) as a single sine function \(y = k \sin(t + \phi)\). You need to calculate \(k\) and \(\phi\).

Step 2: Find Combined Function

Calculate the combined sound function by summing up the two given functions: \[f(t) = f_1(t) + f_2(t) = 5 \sin t + 5 \cos t\]

Step 3: Express as Single Sine Function

Recognize that \(f(t) = 5 \sin t + 5 \cos t\) should resemble \(k \sin(t + \phi)\), which can be rewritten using the sine addition formula: \[k \sin(t + \phi) = k (\sin t \cos \phi + \cos t \sin \phi)\]

Step 4: Equate Coefficients

Equate the coefficients from \(5 \sin t + 5 \cos t\) with \(k(\sin t \cos \phi + \cos t \sin \phi)\). This gives two equations: \[k \cos \phi = 5\] and \[k \sin \phi = 5\].

Step 5: Solve for k

Square both equations and add: \((k \cos \phi)^2 + (k \sin \phi)^2 = 5^2 + 5^2\), which simplifies to \(k^2 = 50\). Thus, \(k = \sqrt{50} = 5\sqrt{2}\).

Step 6: Solve for phi

Use the quotient of the cosine and sine equations: \[\tan \phi = \frac{k \sin \phi}{k \cos \phi} = \frac{5}{5} = 1\]. Therefore, \(\phi = \frac{\pi}{4}\) radians.

Key Concepts

Amplitude and Phase Transformation

Amplitude and phase transformations are essential when working with trigonometric functions. These transformations help in converting a combination of sine and cosine waves into a standard sine wave form.

In the problem, the given functions are combined into one:

• The amplitude transformation involves finding the amplitude of the resultant wave, denoted as \(k\). This is the maximum value the wave reaches, and can be calculated through the Pythagorean relationship: \(k = \sqrt{(a^2 + b^2)}\), where \(a\) and \(b\) are the coefficients of the sine and cosine components.
• The phase shift transformation, denoted as \(\phi\), indicates how much the wave is horizontally shifted on the graph. To find \(\phi\), the formula \(\tan \phi = \frac{b}{a}\) is used.

These transformations help convert the signal into a sine format, making it easier to analyze and graph. For instance, the transformations resulted in \(k = 5\sqrt{2}\) and \(\phi = \frac{\pi}{4}\) in this step by step solution, respectively.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as sine and cosine, is crucial for visualizing how these functions behave over time or space.

The basic shape and characteristics of the graph of a sine function can be determined by:

• The amplitude \(k\), which controls the height of the wave.
• The phase shift \(\phi\), which shows how much the graph is shifted left or right.
• The sine wave's period, which is indicative of how long it takes for the function to complete one full cycle. In a standard sine function \(y = \sin(t)\), the period is \(2\pi\).

By graphing \(y = 5\sin(t) + 5\cos(t)\) as \(y = 5\sqrt{2}\sin(t + \frac{\pi}{4})\), you create a simplified sine curve with a clearer understanding of its transformations. The graph of this equation will have an amplitude of \(5\sqrt{2}\), indicating its maximum reach, and will be shifted by \(\frac{\pi}{4}\) to the left or right. Understanding these components simplifies the representation of complex waveforms.

Sine and Cosine Identity

The sine and cosine identities are pivotal in simplifying trigonometric expressions. They are fundamental in rewriting expressions to understand their underlying waveforms.

Some key identities involved are:

• The Pythagorean identity states that \(\sin^2(t) + \cos^2(t) = 1\), which helps relate sine and cosine to their respective amplitudes effectively.
• The sine addition formula, \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\), enables the conversion of sums of trigonometric terms into single sine or cosine expressions.

In this exercise, using the sine addition formula helps express the function \(5\sin t + 5\cos t\) as \(k \sin(t + \phi)\), where \(k\) and \(\phi\) are effectively derived from the relationships between the individual trigonometric components. The concept provides a bridge to simplify complex trigonometric forms into their base wave functions.