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

The expressions are equivalent if \(A = C \cos \phi\) and \(B = C \sin \phi\), thus \(A^2 + B^2 = C^2\) and \(\tan \phi = B/A\).

Step 1: Expand the Sine Angle Sum

Start by using the angle sum identity for sine. The identity is given by: \(\sin(a + b) = \sin a \cos b + \cos a \sin b.\)Applying this to \(\sin(\omega t + \phi)\), we have:\(\sin(\omega t + \phi) = \sin(\omega t)\cos(\phi) + \cos(\omega t)\sin(\phi).\)Thus, the expression \(C \sin(\omega t + \phi)\) becomes:\(C (\sin(\omega t)\cos(\phi) + \cos(\omega t)\sin(\phi)) = C \cos(\phi) \sin(\omega t) + C \sin(\phi) \cos(\omega t).\)

Step 2: Equating Terms

Equate \(A \sin(\omega t) + B \cos(\omega t)\) with the expanded form from Step 1:\(A \sin(\omega t) + B \cos(\omega t) = C \cos(\phi) \sin(\omega t) + C \sin(\phi) \cos(\omega t).\)From this equality, we deduce:\(A = C \cos(\phi), \quad B = C \sin(\phi).\)

Step 3: Derive Relationship Between A, B, and C

Using the equations \(A = C \cos(\phi)\) and \(B = C \sin(\phi)\), we square both sides:\(A^2 = C^2 \cos^2(\phi)\) and \(B^2 = C^2 \sin^2(\phi)\).Adding these:\(A^2 + B^2 = C^2 \cos^2(\phi) + C^2 \sin^2(\phi) = C^2(\cos^2(\phi) + \sin^2(\phi)) = C^2.\)

Step 4: Show Trigonometric Identity for \(\tan\phi\)

Using the relationships \(A = C \cos(\phi)\) and \(B = C \sin(\phi)\), we find \(\tan(\phi)\) by dividing \(B\) by \(A\):\(\tan(\phi) = \frac{B}{A} = \frac{C \sin(\phi)}{C \cos(\phi)} = \frac{\sin(\phi)}{\cos(\phi)}.\)Thus, \(\tan(\phi) = \frac{B}{A}\).

Step 5: Generalize the Result

Generalize the result to include multiple sin terms:For \(A_1 \sin(\omega t + \phi_1) + A_2 \sin(\omega t + \phi_2) + A_3 \sin(\omega t + \phi_3)\),each sine term can be expressed as a combination of sine and cosine terms using the angle sum identity: \(A_i \sin(\omega t + \phi_i) = A_i(\cos(\phi_i) \sin(\omega t) + \sin(\phi_i) \cos(\omega t)).\)Therefore, the entire expression can be rewritten as:\((\Sigma A_i \cos(\phi_i)) \sin(\omega t) + (\Sigma A_i \sin(\phi_i)) \cos(\omega t).\)

Step 6: Discuss Importance of Identity

This identity shows how any linear combination of sine and cosine waves of the same frequency can be represented as a single sine wave with adjusted amplitude and phase. It is essential in signal processing, simplifying the analysis of oscillations and waves, and ensures that the amplitude \(C\) satisfies \(|C| \geq \max(|A|, |B|)\). This identity applies only when working with linear combinations of sine and cosine terms at the same frequency.

Key Concepts

Sine and Cosine Relationships

Understanding the relationship between sine and cosine is fundamental in trigonometry. These two functions are not just mere waves; they are interconnected by trigonometric identities that allow us to express one in terms of the other in different contexts. This relationship is evident in the sum of angles formula:

• \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]

By using this identity, we can expand expressions such as \( \sin(\omega t + \phi) \) into a form that reveals how sine and cosine come together. When broken down, this expansion shows:- \( \sin(\omega t + \phi) = \sin(\omega t)\cos(\phi) + \cos(\omega t)\sin(\phi) \)This expanded form demonstrates that any combination of these two trigonometric functions can be expressed through their mutual interactions. Such relationships are useful in simplifying complex trigonometric expressions and in solving equations involving sine and cosine.
Distilling complex interactions into simplified equations helps in various applications, from oscillations in physics to signal processing in engineering.

Amplitude and Phase

Amplitude and phase are key characteristics of oscillating systems, like trigonometric functions, which define the behavior of waves. Amplitude refers to the maximum value of the wave, indicating how high (or low) the wave peaks.

• For the expression \( C \sin(\omega t + \phi) \), \( C \) represents the amplitude.

Phase, on the other hand, specifies the starting point or offset of the function in its cycle. It denotes how much the wave shifts horizontally, altering when it appears to start or peak.When dealing with trigonometric functions such as \( A \sin(\omega t) + B \cos(\omega t) \), the amplitude \( C \) and phase \( \phi \) can adjust these sine and cosine combinations into a unified form \( C \sin(\omega t + \phi) \). This transformation is possible because:- \( A = C \cos(\phi) \)- \( B = C \sin(\phi) \)Calculating the amplitude requires understanding the relationship \( A^2 + B^2 = C^2 \), ensuring that:

• The combined wave has maximum amplitude that satisfies \(|C| \geq \max(|A|, |B|)\).

For finding phase, the relationship \( \tan(\phi) = \frac{B}{A} \) is used to determine the angle \( \phi \) through inverse trigonometry.

Linear Combinations of Trigonometric Functions

When discussing linear combinations of trigonometric functions, we delve into how different sine and cosine terms can be summed or subtracted to form new functions. A linear combination involves adding multiples of these basic trigonometric functions, usually of the same frequency:

• For example, \( A \sin(\omega t) + B \cos(\omega t) \) is a linear combination.

Such combinations are pivotal in analyzing and simplifying periodic behavior in mathematics and physics. This approach allows many complex waveforms to be expressed succinctly as a single sinusoidal wave with the same frequency.By transforming this combination into \( C \sin(\omega t + \phi) \), where the expressions for \( A \) and \( B \) align with \( A = C \cos(\phi) \) and \( B = C \sin(\phi) \), a single unified wave is attained. This not only makes mathematical expressions simpler but also aids in practical applications like signal processing, where extracting or modifying signals is crucial.Moreover, understanding this identity is vital for recognizing ways to manipulate signals without changing their original nature, facilitating work in fields such as engineering and physics where wave behaviours intersect with real-world applications.