Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

We now explore the relationship between \(A \sin (\omega t)+\) \(B \cos (\omega t)\) and \(C \sin (\omega t+\phi)\). (a) By expanding \(\sin (\omega t+\phi)\) using the sum of the angles formula, show that the two expressions are equivalent if \(A=C \cos \phi\) and \(B=C \sin \phi .\) (b) Consequently, show that \(A^{2}+B^{2}=C^{2}\) and that \(\phi\) then satisfies the equation \(\tan \phi=\frac{B}{A}\). (c) Generalize your result to state a proposition about \(A_{1} \sin \left(\omega t+\phi_{1}\right)+A_{2} \sin \left(\omega t+\phi_{2}\right)+A_{3} \sin \left(\omega t+\phi_{3}\right)\). (d) Write an essay, in your own words, that expresses the importance of the identity between \(A \sin (\omega t)+B \cos (\omega t)\) and \(C \sin (\omega t+\phi) .\) Be sure to note that \(|C| \geq \max (|A|,|B|)\) and that the identity holds only when you are forming a linear combination (adding and/or subtracting multiples of single powers) of sine and cosine of the same frequency.

Step-by-Step Solution

Answer

(a) Use sum of angles to show equivalence for given conditions. (b) Show \( A^2 + B^2 = C^2 \) and \( \tan \phi = \frac{B}{A} \). (c) Extend the logic to multiple sine terms. (d) The identity clarifies complex harmonics and phase relationships.

Step 1: Expand the Sine with Sum of Angles Formula

The given identity involves comparing two trigonometric expressions. We start by expanding \( \sin(\omega t + \phi) \) using the sum of angles formula: \[ \sin(\omega t + \phi) = \sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi) \] Substitute this expansion into the expression for \( C \sin(\omega t + \phi) \): \[ C \sin(\omega t + \phi) = C (\sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi)) \] This expands to: \[ C \cos(\phi) \sin(\omega t) + C \sin(\phi) \cos(\omega t) \] This must be equivalent to the expression \( A \sin(\omega t) + B \cos(\omega t) \). Therefore, \( A = C \cos(\phi) \) and \( B = C \sin(\phi) \).

Step 2: Prove Relationship Between A, B, and C

With \( A = C \cos(\phi) \) and \( B = C \sin(\phi) \), square each and add them: \[ A^2 = (C \cos(\phi))^2, \quad B^2 = (C \sin(\phi))^2 \] Thus, \[ A^2 + B^2 = (C \cos(\phi))^2 + (C \sin(\phi))^2 \] Utilize the Pythagorean identity \( \cos^2(\phi) + \sin^2(\phi) = 1 \):\[ A^2 + B^2 = C^2 (\cos^2(\phi) + \sin^2(\phi)) = C^2 \]Therefore, \( A^2 + B^2 = C^2 \).

Step 3: Establish Relationship for φ Using Tangent

To find the angle \( \phi \), consider the ratio of \( B \) and \( A \):\[ \tan(\phi) = \frac{\sin(\phi)}{\cos(\phi)} = \frac{B/C}{A/C} = \frac{B}{A} \]This verifies that \( \phi \) satisfies the equation \( \tan(\phi) = \frac{B}{A} \).

Step 4: Generalize the Proposition

Consider a sum \( A_1 \sin(\omega t + \phi_1) + A_2 \sin(\omega t + \phi_2) + A_3 \sin(\omega t + \phi_3) \). Each term can be rewritten similarly using the sum of angles identity. We can express each as \( A_i \sin(\omega t + \phi_i) = A_i \cos(\phi_i) \sin(\omega t) + A_i \sin(\phi_i) \cos(\omega t) \).By adding these, the overall expression remains a linear combination of \( \sin(\omega t) \) and \( \cos(\omega t) \) and can be combined into a single sine function with an overall constant \( C \) and phase \( \phi \).

Step 5: Discuss the Importance of the Identity

The identity \( A \sin(\omega t) + B \cos(\omega t) = C \sin(\omega t + \phi) \) is critical in harmonics and signal processing. It allows for expressing complex harmonic motion in the more interpretable sine form, clarifying phase relationships. A key concept is that \(|C| \geq \max(|A|,|B|)\), ensuring that \( C \) encompasses both amplitude contributions from \( A \) and \( B \). This condition applies specifically when working with components of the same frequency. Understanding this can simplify analysis and synthesis in various fields such as electronics, acoustics, and communications.

Key Concepts

Sum of Angles Formula

The Sum of Angles Formula is a vital tool in trigonometry, allowing us to break down complex trigonometric expressions into more manageable parts. Specifically, for the sine function, it is expressed as: \[ \sin(\omega t + \phi) = \sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi) \] This formula helps unravel the expression for a sine function where the angle is a sum of two parts. It reveals how two trigonometric components, sine and cosine, of the separate angles, combine to give the sine of their sum.

• Helps in equating expressions with trigonometric components.
• Facilitates the comparison of different forms of trigonometric expressions.
• Essential in verifying the equivalency between trigonometric identities.

In this exercise, using the sum of angles formula allowed us to transform a complex trigonometric expression into an equivalent form. This transformation is essential for proving the equivalency between different trigonometric forms and showcases the power of this formula. When given an equation involving sums or differences, knowing this identity allows you to solve or simplify the equation effectively.

Pythagorean Identity

The Pythagorean Identity is another cornerstone of trigonometry, which directly stems from the Pythagorean theorem. The identity is represented by the equation:\[ \cos^2(\phi) + \sin^2(\phi) = 1 \]This identity is crucial when dealing with relationships between sine and cosine functions.

• Ensures a foundational relationship between sine and cosine.
• Helps in deriving other trigonometric identities.
• Used to simplify expressions involving these functions.

In this context, we use the Pythagorean Identity to establish that: \[ A^2 + B^2 = C^2 \]By recognizing that \(A\) and \(B\) can be expressed as proportions of \(C\), the Pythagorean Identity makes it possible to derive such an important relationship. This relates directly to how we combine sine and cosine functions in trigonometry:

• Proves that the resultant amplitude \(C\) accounts for contributions from \(A\) and \(B\).
• Ensures that any linear combination of sine and cosine remains consistent within a circle.

This identity underscores how trigonometric functions interplay within a circular dynamic, facilitating the understanding of harmonic motion and wave phenomena.

Harmonic Motion

Harmonic motion refers to the repetitive, oscillating motion that occurs in systems such as springs, pendulums, and waves. In mathematics, it can be described using trigonometric functions like sine and cosine.

• Describes systems that move back and forth in a regular pattern.
• Simplified using trigonometric identities.
• Key in physics for understanding mechanical vibrations, waves, and acoustics.

In the given exercise, the expression \(A \sin(\omega t) + B \cos(\omega t)\) models such motion. When harmonic motion is expressed in terms of just one sine function with phase, such as \(C \sin(\omega t + \phi)\), it's easier to identify the motion's characteristics:

• The amplitude \(C\) and phase \(\phi\) are directly observable.
• Understanding phase shifts becomes more straightforward.
• Enables a clearer interpretation of the combined effects of the sine and cosine terms.

Recognizing and utilizing the equivalent forms of these expressions aids in areas such as acoustics and signal processing where harmonic motions are prevalent.

Signal Processing

Signal processing is a branch of engineering focused on analyzing, modifying, and synthesizing signals such as sound, images, and scientific measurements. Trigonometric identities are vital tools in this field.

• Involves the manipulation of signal properties like frequency, phase, and amplitude.
• Uses trigonometric forms to facilitate transformations and interpretations.
• Important in digital communications, audio engineering, and image processing.

The identity \(A \sin(\omega t) + B \cos(\omega t) = C \sin(\omega t + \phi)\) simplifies the processing of signals with sine and cosine components by translating them into a single sine function. Here’s how this benefits signal processing:

• Consolidates multiple frequency components into a single interpretable form.
• Helps in assessing the phase and amplitude more efficiently.
• Enhances the ability to filter, mix, and modulate signals accurately.

This means more effective techniques in digital signal processing, such as Fast Fourier Transforms (FFT) and signal filtering, thereby aiding in practical applications like sound compression and noise reduction.