Problem 136 - Precalculus

Question

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

Step-by-Step Solution

Answer

The amplitude of the function is 2.

Step 1: Understanding Amplitude

The amplitude of a function like \(a \cos x + b \sin x\) can be determined by the formula \(\sqrt{a^2 + b^2}\). Here \(a = \sqrt{3}\) and \(b = -1\). The amplitude refers to the maximum value of the wave above or below the horizontal axis.

Step 2: Calculating the Amplitude

To find the amplitude, use the formula \(\sqrt{a^2 + b^2}\). Substitute \(a = \sqrt{3}\) and \(b = -1\) into the formula. Therefore, the amplitude is \(\sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2\). The amplitude of the function is 2.

Step 3: Visual Verification with Graphing

Use a graphing calculator to verify. Enter \(Y_1 = \sqrt{3}\cos x\), \(Y_2 = \sin x\), and \(Y_3 = \sqrt{3}\cos x - \sin x\) to graph these functions in the same viewing window. You will see that the peaks and troughs of \(Y_3\) reach a maximum distance of 2 from the centerline, confirming the amplitude.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding periodic phenomena such as waves and oscillations. The most common are sine (\( \sin x \)), cosine (\( \cos x \)), and tangent (\( \tan x \)) functions. These functions are crucial in fields such as physics, engineering, and even music and visual arts.
Each function has unique properties:

• The sine and cosine functions have a period of \( 2\pi \), which means they repeat every \( 2\pi \) radians.
• The amplitude of a sine or cosine function is the height from the centerline to the peak, determining how "tall" the wave appears.
• These trigonometric functions are continuous and smooth, offering a clear depiction of natural phenomena.

When you have combined functions, like \( a \cos x + b \sin x \), you can calculate the amplitude using the formula: \( \sqrt{a^2 + b^2} \). This helps in determining how varied the function becomes from its central axis. For example, in the given function \( y = \sqrt{3} \cos x - \sin x \), the amplitude calculation helps understand the wave's peak elevation from its baseline.

Graphing Calculator

Graphing calculators provide an intuitive way to visualize mathematical functions, making it easier to understand complex concepts. By plotting multiple functions on the same graph, you can observe patterns and interactions between different mathematical phenomena.
In the given exercise, students use a graphing calculator to plot three functions:

• \( Y_1 = \sqrt{3} \cos x \)
• \( Y_2 = \sin x \)
• \( Y_3 = \sqrt{3} \cos x - \sin x \)

Using the graphing calculator, you can see how \( Y_3 \) combines elements from both \( Y_1 \) and \( Y_2 \), and how its waveform exceeds the individual components. By setting a common viewing window, you can verify statistical results such as the peak values, amplitude, and phase shifts visually. This capability highlights the power of technology in reinforcing mathematical concepts and enhances comprehension through visual proof.

Function Analysis

Function analysis is a process that helps in understanding and interpreting mathematical expressions or equations. It involves identifying the characteristics of a function, such as its amplitude, period, and phase shift.
Analysis begins with recognizing the type of function you are dealing with. For trigonometric functions, we look for how these functions behave over their domain. The amplitude tells us the vertical stretch, while the period tells us how long it takes for the pattern to repeat.
In this exercise, once we have computed the amplitude using \( \sqrt{a^2 + b^2} \), the next step is to validate our result analytically and visually. This involves:

• Checking that the peaks and troughs of the wave on the graph reach \( \pm 2 \), confirming the calculated amplitude.
• Examining how the components \( \sqrt{3} \cos x \) and \( -\sin x \) interact to affect the resultant wave \( y \).

By conducting such an analysis, students gain a deeper insight into how different mathematical functions influence each other and how to interpret real-world phenomena through mathematics.