Problem 124 - Precalculus

Question

Can the \(y\) -coordinate of a point on the graph of \(y=A \sin B x+3 A \cos \left(\frac{B}{2} x\right)\) exceed \(4 A ?\) Explain. (Assume that \(A > 0 .)\)

Step-by-Step Solution

Answer

No, the maximum y-coordinate cannot exceed 4A.

Step 1: Identify the Function Components

The given function is \( y = A \sin(Bx) + 3A \cos \left( \frac{B}{2} x \right) \). The function is composed of two trigonometric components: \( A \sin(Bx) \) and \( 3A \cos \left( \frac{B}{2} x \right) \). Both components have amplitudes because of the coefficients multiplying the sine and cosine functions.

Step 2: Determine the Amplitude of Each Component

The amplitude of \( A \sin(Bx) \) is \( A \), as the sine function varies from -1 to 1. The amplitude of \( 3A \cos \left( \frac{B}{2} x \right) \) is \( 3A \), since the cosine function also varies from -1 to 1. Thus, when each function is at maximum, \( A \sin(Bx) = A \) and \( 3A \cos \left( \frac{B}{2} x \right) = 3A \).

Step 3: Calculate the Maximum Value of the Function

When both components reach their maximum values simultaneously (assuming the same or appropriate phase), the maximum value of \( y \) would be \( A + 3A = 4A \). However, the maximum value occurs theoretically when sine and cosine align appropriately. Typically, absolute sine and cosine are difficult to align perfectly due to their periodic nature.

Step 4: Assess the Possibility of Exceeding 4A

The theoretical maximum value for \( y \, \) given the amplitudes, reaches 4A. This value occurs in an ideal scenario where both trigonometric functions add constructively. In practical terms, mixtures of sine and cosine like this rarely exceed combined maximum due to phase shifts and periodic out-of-phase points.

Step 5: Conclusion on Maximum Value

Ultimately, the calculated combined maximum achievable under ideal conditions is 4A. However, the maximum value cannot exceed 4A because both functions cannot be simultaneously beyond their respective maximum contributions without changing coefficients, which contradicts the problem's context.

Key Concepts

Amplitude

Amplitude in trigonometry is a measure of how far a wave can 'rise' or 'fall' from its middle point, commonly known as the equilibrium position. In simpler terms, it's the height of the wave.

• The amplitude of a sine or cosine function is determined by the coefficient in front of the function, such as in the function \( y = A \sin(Bx) \), where the amplitude is \( A \).
• Amplitudes can be understood better with a graph: the graph of \( y = A \sin(Bx) \) will oscillate between \( A \) and \( -A \).
• When combined like in \( y = A \sin(Bx) + 3A \cos(\frac{B}{2}x) \), each component's amplitude contributes to the overall behavior of the graph.

If you have two components with amplitudes \( A \) and \( 3A \), the maximum potential contribution would sum them due to their sine and cosine nature, explaining why it could reach up to \( 4A \) but not beyond.

Sine Function

The sine function is one of the basic trigonometric functions that relates to an angle in a right triangle. It is periodic, meaning it repeats its values over regularly spaced intervals.

• In mathematical terms, the sine of an angle \( \theta \) in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
• The general form of the sine function is \( y = A \sin(Bx + C) \), where \( A \) is the amplitude, \( B \) affects the period, and \( C \) represents a phase shift.
• This function oscillates smoothly between -1 and 1, multiplied by the amplitude \( A \), the function ranges between \( -A \) and \( A \).

In the given expression, \( y = A \sin(Bx) \) denotes this very sine wave pattern multiplied by a factor \( A \), with its contribution essential to understanding the maximum value when combined with other trigonometric expressions.

Cosine Function

The cosine function is another fundamental trigonometric function. Similar to sine, it embodies cyclical patterns and is paramount in waveform analysis.

• Like the sine, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
• The cosine function is expressed as \( y = A \cos(Bx + C) \), describing a wave like sine but starting at its maximum when \( x = 0 \).
• The standard cosine curve spans from -1 to 1. When amplified by \( A \), it varies from \( -A \) to \( A \).

In the function \( 3A \cos(\frac{B}{2}x) \), the presence of "\( 3A \)" means the wave's amplitude is scaled up, reaching from "\(-3A\)" to "\(3A\)". This scaling factor significantly impacts the combined outcomes of sine and cosine components in equations involving both, illustrating why, when theoretically all conditions align, it can contribute up to the maximum 3A, explaining jointly their peak 4A potential.