Problem 83 - Precalculus

Question

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

Step-by-Step Solution

Answer

The amplitude is \(\sqrt{2}\).

Step 1: Understand the Problem

We need to find the amplitude of the function \(y = \cos x + \sin x\). Amplitude refers to the maximum deviation from the average value, or zero line, for periodic functions. Here, both \(\cos x\) and \(\sin x\) are periodic functions with amplitudes of 1.

Step 2: Rewrite the Combined Function

To simplify \(y = \cos x + \sin x\), we use the identity for a combination of sinusoidal functions: any function of the form \(A\cos x + B\sin x\) can be rewritten as \(R \cos(x - \phi)\). Here, \(A = 1\) and \(B = 1\).

Step 3: Calculate the Resultant Amplitude

The amplitude \(R\) of the function in form \(A\cos x + B\sin x\) is calculated as \(R = \sqrt{A^2 + B^2}\). Substituting \(A = 1\) and \(B = 1\), we get: \[ R = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414 \]

Step 4: Graphing the Functions

Using a graphing calculator, plot \(Y_1 = \cos x\), \(Y_2 = \sin x\), and \(Y_3 = \cos x + \sin x\) in the same window. Observe how the graph of \(\cos x + \sin x\) behaves compared to \(\cos x\) and \(\sin x\). This visual helps confirm that the amplitude is approximately \(\sqrt{2}\).

Step 5: Verify with a Graphing Tool

After plotting, you'll see the wave of \(\cos x + \sin x\) achieves maximum and minimum values that reach about \(\pm \sqrt{2}\). This confirms that the calculated amplitude is correct.

Key Concepts

Periodic Functions

A periodic function is a function that repeats its values in regular intervals or periods. The most common examples of periodic functions are trigonometric functions like sine and cosine.
They have specific properties that make them predictable and easy to study.

• Definition: A function is periodic if there exists a positive number \(T\) such that \(f(x + T) = f(x)\) for all values of \(x\) in the domain of \(f\). The smallest such positive \(T\) is called the period of the function.
• Properties: The graphs of sine and cosine functions exhibit wave-like patterns repeated every \(2\pi\) radians.

Understanding the periodic nature of functions like \(\cos x\) and \(\sin x\) allows us to deduce their behaviors over time. When combined, as in \(y = \cos x + \sin x\), they can create new functions that still retain periodic characteristics and allow us to study their amplitudes and periods.

Graphing Calculator

A graphing calculator is an essential tool to visualize mathematical functions, especially when dealing with trigonometric functions. When plotting \(y = \cos x\), \(\sin x\), and \(\cos x + \sin x\), a graphing calculator helps us comprehend their interactions and visualize the resultant function's behavior.

• Function Plots: You can enter functions like \(Y_1 = \cos x\), \(Y_2 = \sin x\), and \(Y_3 = \cos x + \sin x\) to see how they appear individually and in combination.
• Amplitude Visual Check: By graphing \(\cos x + \sin x\), you can visually confirm the maximum values reach approximately \(\sqrt{2}\), consistent with the amplitude calculation.

Using the graphing calculator simplifies the process of studying transformations and combinations of periodic functions. It provides an immediate visual representation of complex equations, aiding conceptual understanding.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the equation are defined. They are incredibly useful when simplifying expressions or solving trigonometric equations.

Key Identities

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \).
• Sum-to-Product Formulas: These enable rewriting combinations of sine and cosine, such as \(A \cos x + B \sin x\), into a single trigonometric function: \( R \cos(x - \phi) \), where \(R = \sqrt{A^2 + B^2}\).

In the problem, we used the sum-to-product formula to transform the given function \(y = \cos x + \sin x\) into a more manageable form, allowing us to easily determine the amplitude. Recognizing these identities and using them effectively is key in trigonometry.