Problem 51 - Calculus Single Variable

Question

Suppose that \(A\) and \(B\) are constants. Use the Addition Formula for the sine to find an amplitude \(C\) and a phase shift \(\phi\) (both in terms of \(A\) and \(B\) ) such that $$A \cos (\theta)+B \sin (\theta)=C \sin (\theta+\phi)$$ for every \(\theta\).

Step-by-Step Solution

Answer

The amplitude \( C = \sqrt{A^2 + B^2} \) and phase shift \( \phi = \tan^{-1}(\frac{A}{B}) \).

Step 1: Express the Right-Hand Side

The expression on the right-hand side is given as \( C \sin(\theta + \phi) \). Using the sine addition formula, this can be expanded as \( C(\sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi)) \).

Step 2: Match Coefficients

Ensure the coefficients of \( \cos(\theta) \) and \( \sin(\theta) \) match on both sides of the equation. This gives us two equations: \( A = C\sin(\phi) \) and \( B = C\cos(\phi) \).

Step 3: Solve for Amplitude C

To find \( C \), square both equations derived in the matching step and add them: \( A^2 + B^2 = (C\sin(\phi))^2 + (C\cos(\phi))^2\). Simplifying using the Pythagorean identity, \( C^2(\sin^2(\phi) + \cos^2(\phi)) = C^2 \), yields \( C = \sqrt{A^2 + B^2} \).

Step 4: Solve for Phase Shift \( \phi \)

Divide the two equations found in Step 2 to solve for \( \phi \): \( \frac{A}{B} = \frac{C\sin(\phi)}{C\cos(\phi)} = \tan(\phi) \). Then, \( \phi = \tan^{-1}(\frac{A}{B}) \).

Step 5: Verify the Solution

Substitute \( C = \sqrt{A^2 + B^2} \) and \( \phi = \tan^{-1}(\frac{A}{B}) \) back into \( C\sin(\theta+\phi) \) and confirm it simplifies to \( A \cos(\theta)+B \sin(\theta) \). This verification ensures that the derived values satisfy the original equation.

Key Concepts

Amplitude

The amplitude of a trigonometric function is a crucial concept in understanding the behavior of sine and cosine functions. It represents the peak value or the strength of the wave-like graph. Simply put, amplitude tells us how tall or deep the wave reaches relative to its central axis.
In mathematical terms, for the expression \( A \cos(\theta) + B \sin(\theta) = C \sin(\theta + \phi) \), the amplitude \( C \) is found using the formula \( C = \sqrt{A^2 + B^2} \). This formula stems from the Pythagorean identity, which states that the sum of the squares of the sine and cosine of an angle equals one.
Thus, to determine the amplitude in this context:

• Square each coefficient \( A \) and \( B \).
• Add these squares together to utilize the Pythagorean theorem.
• Take the square root of this sum to find the amplitude \( C \).

This calculated amplitude will control how far the function stretches vertically from its center.

Phase Shift

Phase shift is a key concept in trigonometry that describes the horizontal movement of the graph of a sine or cosine function. It indicates how much the entire waveform is shifted left or right. Understanding phase shift is essential when manipulating trigonometric graphs to model real-world phenomena.
In the expression \( A \cos(\theta) + B \sin(\theta) = C \sin(\theta + \phi) \), the phase shift \( \phi \) can be determined by solving the equation \( \tan(\phi) = \frac{A}{B} \).

• First, use the inset relationship \( \tan(\phi) \), which comes from matching coefficients on both sides of the equation.
• Using the inverse tangent function, \( \phi = \tan^{-1}(\frac{A}{B}) \), solve for the phase shift.

This shift directly affects where the sine graph begins on the x-axis, changing the function's starting point without altering its shape or amplitude.

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving trigonometric expressions and equations. These are equations that are true for all values of the variable within their domain. The Pythagorean identity is one of the most fundamental identities, given as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
This identity is instrumental in deriving the amplitude and phase shift formulas. In our context, we used another identity, the addition formula for sine: \[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]

• This formula helped express the right-hand side of the initial equation \( C \sin(\theta + \phi) \) and equate it with \( A \cos(\theta) + B \sin(\theta) \).
• Consequently, it enabled us to focus on matching coefficients to find expressions for amplitude and phase shift.

Understanding these identities allows for the manipulation and transformation of trigonometric functions, vital for solving complex problems.

Sine and Cosine Functions

Sine and cosine functions are the building blocks of trigonometry, describing oscillations and waves. Both are periodic functions with specific properties that make them suited for depicting cyclical behavior.

• Sine Function: Typically starts at 0, reaches a maximum at 90 degrees (or \( \pi/2 \)), goes to 0 at 180 degrees (or \( \pi \)), and reaches a minimum at 270 degrees (or \( 3\pi/2 \)).
• Cosine Function: Starts at 1, decreases to 0 at 90 degrees (or \( \pi/2 \)), reaches -1 at 180 degrees, and returns to 0 at 270 degrees (or \( 3\pi/2 \)).

When transforming these functions into each other through phase shifts, we utilize their inherent properties and relationships. For instance, the cosine function can be seen as a sine function that is phase-shifted by 90 degrees. These functions form the basis of the exercise where we manipulate their sums using the addition formula to derive equivalent forms in terms of amplitude and phase shift. Such manipulations are fundamental in applications such as signal processing and wave analysis.