Problem 135 - Precalculus

Question

What is the amplitude of the function \(y=3 \cos x+4 \sin x ?\) Use a graphing calculator to graph \(Y_{1}=3 \cos x, Y_{2}=4 \sin x\) and \(Y_{3}=3 \cos x+4 \sin x\) in the same viewing window.

Step-by-Step Solution

Answer

The amplitude is 5.

Step 1: Understanding the Question

We need to determine the amplitude of the function \( y = 3 \cos x + 4 \sin x \). The amplitude of a trigonometric function of the form \( a \cos x + b \sin x \) can be calculated using the formula \( \sqrt{a^2 + b^2} \). In this case, \( a = 3 \) and \( b = 4 \).

Step 2: Calculating the Amplitude

To find the amplitude of \( y = 3 \cos x + 4 \sin x \), apply the formula \( \sqrt{a^2 + b^2} \). This becomes \( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). So, the amplitude is 5.

Step 3: Visualizing the Functions

To confirm our calculation, use a graphing calculator. First, graph \( Y_1 = 3 \cos x \) and \( Y_2 = 4 \sin x \). Then, graph \( Y_3 = 3 \cos x + 4 \sin x \) in the same window. This will show how the combined wave's maximum height aligns with our calculated amplitude.

Step 4: Interpreting the Graph

By observing the graph of \( Y_3 \), note that the maximum value reaches 5, confirming our calculation. You may see the graph peaking at 5 and dipping to -5, which demonstrates the amplitude is indeed 5.

Key Concepts

Amplitude

In trigonometry, the amplitude of a function refers to the maximum distance it reaches from its central or equilibrium position. It can be thought of as the height of the wave from the center line to the peak (or trough).
A trigonometric function's amplitude is often discussed in the context of sine and cosine functions. For a function of the form \( a \cos x + b \sin x \), the amplitude is calculated using the formula \( \sqrt{a^2 + b^2} \). This allows you to determine how "tall" the wave is.

• In the specific function \( y = 3 \cos x + 4 \sin x \), we use the formula mentioned to find the amplitude.
• Plugging the values, \( a = 3 \) and \( b = 4 \), into the formula gives \( \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \).
• Thus, the amplitude is 5. This tells us that the graph will extend 5 units above and below the center line.

This concept is crucial in understanding the wave's behavior because the amplitude reveals how intense or severe the oscillations will be.

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse.
This function creates a wave-like graph that oscillates between -1 and 1 naturally without any adjustments.

• The standard form of a cosine wave is \( y = a \cos x \), where \( a \) represents the amplitude.
• In our scenario, \( y = 3 \cos x \) indicates an amplitude of 3, meaning the wave stretches 3 units from the center line.

Aside from its mathematical definition, cosine curves are essential in modeling various physical phenomena such as sound waves and light waves.
When combined with other functions like sine in our equation, the effect of cosine's amplitude contributes to the overall wave's shape and size, increasing its richness and complexity.

Sine Function

The sine function, expressed as \( \sin x \), is another key trigonometric function defined as the ratio of the opposite side to the hypotenuse in a right triangle. Like cosine, the sine function produces a periodic wave that oscillates naturally between -1 and 1.

For a standard sine wave equation \( y = b \sin x \), the amplitude is determined by the coefficient \( b \).

• In the given equation, \( y = 4 \sin x \), the amplitude is 4, which means the wave reaches 4 units from its central axis.
• Similar to the cosine graph, the sine graph plays a crucial role in various scientific calculations and modeling processes, such as signal processing and vibration analysis.

When you combine sine and cosine functions, like in our function \( y = 3 \cos x + 4 \sin x \), their respective amplitudes and phases can dramatically alter the resulting wave's appearance and behavior. Understanding each function's role is essential in accurately predicting and interpreting these combined wave patterns.