Problem 99 - Algebra and Trigonometry

Question

Let \(f(x)=\sin 6 x+\sin 7 x\) (a) Graph \(y=f(x)\) (b) Verify that \(f(x)=2 \cos \frac{1}{2} x \sin \frac{13}{2} x\) (c) Graph \(y=2 \cos \frac{1}{2} x\) and \(y=-2 \cos \frac{1}{2} x\) , together with the graph in part \((a),\) in the same viewing rectangle. How are these graphs related to the graph of \(f ?\)

Step-by-Step Solution

Answer

The graphs of \( y = \pm 2 \cos\left(\frac{1}{2}x\right) \) are the envelope functions for \( f(x) = \sin(6x) + \sin(7x) \). They bound the oscillations of the function.

Step 1: Graph y = f(x)

To graph the function \( y = f(x) = \sin(6x) + \sin(7x)\), use a graphing calculator or software like Desmos or GeoGebra. Plot the function over a domain, such as \(-2\pi \leq x \leq 2\pi\), to see the behavior of the function.

Step 2: Verify the Trigonometric Identity

We need to use the sum-to-product identities to verify the given expression: \( f(x) = 2\cos\left(\frac{1}{2}x\right) \sin\left(\frac{13}{2}x\right) \).1. Apply the sum identity for sines: \[ \sin A + \sin B = 2\sin\left( \frac{A+B}{2} \right)\cos\left( \frac{A-B}{2} \right) \] where \( A = 7x \) and \( B = 6x \).2. Calculate \( A+B = 13x \) and \( A-B = x \).3. Substitute these into the identity to get: \[ \sin 6x + \sin 7x = 2\sin\left( \frac{13x}{2} \right)\cos\left( \frac{x}{2} \right) \]4. Thus, the expression is verified as: \[ f(x) = 2\cos\left(\frac{1}{2}x\right)\sin\left(\frac{13}{2}x\right) \]

Step 3: Graph y = 2 cos(1/2 x) and y = -2 cos(1/2 x)

Plot the function \( y = 2\cos\left(\frac{1}{2}x\right) \) and its negative \( y = -2\cos\left(\frac{1}{2}x\right) \) on the same axis as \( f(x) \). Use the same domain \(-2\pi \leq x \leq 2\pi\). Compare these graphs with the graph of \( f(x) = \sin(6x) + \sin(7x) \).

Step 4: Analyze the Relation between Graphs

The graphs of \( y = \pm 2 \cos\left(\frac{1}{2}x\right) \) represent the envelope functions for \( f(x) = \sin(6x) + \sin(7x) \). This means that every peak and trough of the oscillating function \( f(x) \) is bounded by these two cosine functions. Hence, these two cosine graphs form the boundaries within which \( f(x) \) oscillates.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions helps us visualize how they behave over different intervals. For functions like \( y = f(x) = \sin(6x) + \sin(7x) \), graphing shows their periodic nature. To graph this function, you can use graphing tools such as Desmos or GeoGebra.

Start by selecting a suitable domain, like \(-2\pi \leq x \leq 2\pi\). This captures several cycles and provides a clear picture of how the function oscillates.

• Use graphing software to input the function.
• Ensure your chosen domain covers enough cycles to understand patterns.
• Observe the amplitude and frequency changes in comparison to simpler sine functions.

Visualizing the graph helps us understand the interaction between the two sine functions and their combined effect on the waveform.

Sum-to-Product Identities

Sum-to-product identities are powerful tools in trigonometry, simplifying the combination of sine and cosine functions. In this exercise, we explore the identity for sine sums:

\[\sin A + \sin B = 2 \sin\left( \frac{A+B}{2} \right) \cos\left( \frac{A-B}{2} \right)\]

Using this formula, we convert \( \sin(6x) + \sin(7x) \) to:

\[2 \cos\left(\frac{1}{2}x\right) \sin\left(\frac{13}{2}x\right)\]

• Identify \(A = 7x\) and \(B = 6x\) from the original functions.
• Calculate \(A+B = 13x\) and \(A-B = x\).
• Apply these results to simplify the sum using the identity.

This transformation utilizes symmetry properties of sine and cosine, making complex expressions more manageable.

Trigonometric Function Envelopes

Trigonometric function envelopes create boundaries for wave-like graphs. By using envelope functions like \( y = 2 \cos\left( \frac{1}{2}x \right) \) and its negative counterpart, we frame the peaks and troughs of \( f(x) = \sin(6x) + \sin(7x) \). These envelope functions do not oscillate themselves, but indicate the furthest reach of the oscillations.

• Graph \( y = 2 \cos\left( \frac{1}{2}x \right) \) and \( y = -2 \cos\left( \frac{1}{2}x \right) \).
• Notice how these mimic the amplitude limitations of \( f(x) \).
• These provide a visual guide for where \( f(x) \) reaches its maximum and minimum values.

Plotting the main function alongside its envelopes clarifies its range. This method of visualization aids in understanding concepts like amplitude and frequency in oscillating phenomena.