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Magazine Chapter 5: Problem 29
Graph each of the following from \(x=0\) to \(x=2 \pi\). $$y=3 \cos 7 x \cos 5 x+3 \sin 7 x \sin 5 x$$
Short Answer
Expert verified
The function simplifies to \( y = 3 \cos(2x) \) and completes two cycles from \( x = 0 \) to \( 2\pi \).
Step by step solution
01
Apply Trigonometric Identity
Notice that the given function can be simplified using the angle-sum identity for cosine. The identity states that \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). Apply this identity to the expression \( 3 \cos 7x \cos 5x + 3 \sin 7x \sin 5x \).
02
Simplify the Expression
Utilize the identity, where \( A = 7x \) and \( B = 5x \). Substitute into the identity: \[ 3 \cos 7x \cos 5x + 3 \sin 7x \sin 5x = 3 (\cos(7x - 5x)). \] This further simplifies to \( 3 \cos(2x) \).
03
Sketch the Graph of a Cosine Function
Now, we need to graph \( y = 3 \cos(2x) \) from \( x = 0 \) to \( x = 2 \pi \). The cosine function \( \cos(2x) \) has a period of \( \pi \) since the period of \( \cos(kx) \) is \( \frac{2\pi}{k} \). Thus, it completes one full cycle between \( x = 0 \) and \( x = \pi \), and then another from \( x = \pi \) to \( x = 2\pi \).
04
Identify Key Points
Since we know the period is \( \pi \), we mark key points and turning points at multiples of \( \frac{\pi}{4} \). At \( x = 0 \), \( \cos(2 \times 0) = 1 \), so the function starts at 3 (as amplitude is 3). At \( x = \frac{\pi}{4} \), \( \cos(\frac{\pi}{2}) = 0 \), so value is 0. It reaches -3 at \( x = \frac{\pi}{2} \).
05
Plot the Points
Continue plotting key points: back to 0 at \( x = \frac{3\pi}{4} \), 3 again at \( x = \pi \), continuing this way to \( x = 2\pi \). The function oscillates with an amplitude of 3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle-Sum Identity
When handling trigonometric expressions, an important tool is the angle-sum identity. This mathematical identity helps to simplify expressions involving sines and cosines of multiple angles. The specific identity we use for cosine is given by \(\cos(A - B) = \cos A \cos B + \sin A \sin B\).
In our exercise, we apply this identity to the expression \(3 \cos 7x \cos 5x + 3 \sin 7x \sin 5x\). By recognizing that this expression matches the form of the angle-sum identity with \(A = 7x\) and \(B = 5x\), we can simplify the expression.
In our exercise, we apply this identity to the expression \(3 \cos 7x \cos 5x + 3 \sin 7x \sin 5x\). By recognizing that this expression matches the form of the angle-sum identity with \(A = 7x\) and \(B = 5x\), we can simplify the expression.
- First, we combine the terms: \(3 (\cos(7x) \cos(5x) + \sin(7x) \sin(5x))\).
- Next, substituting into the identity, we get: \(3 \cos(7x - 5x)\).
- Simplifying, this becomes \(3 \cos(2x)\).
Cosine Function
The cosine function is one of the fundamental trigonometric functions, integral to understanding wave-like behavior and periodic phenomena. The standard cosine function is denoted as \(\cos(x)\), which oscillates between -1 and 1 as \(x\) varies. For the expression \(3 \cos(2x)\), there are a few modifications to consider.
- Amplitude: The number 3 in \(3 \cos(2x)\) represents the amplitude. It scales the height of the oscillations, so the graph will range between -3 and 3 instead of -1 and 1.
- Frequency: The value 2 inside the cosine function, \(\cos(2x)\), affects the frequency. It means the wave will complete two full cycles over the interval \((0, 2\pi)\).
- Period: For a function \(\cos(kx)\), the period is \(\frac{2\pi}{k}\). Here, \(k = 2\), so the period of the function is \(\pi\), indicating one full cycle from \(x=0\) to \(x=\pi\).
Graphing Techniques
Graphing trigonometric functions requires understanding their behavior and key characteristics. The graph of our expression \(y = 3 \cos(2x)\) can be plotted effectively by considering the amplitude, period, and key points.
To create the graph from \(x = 0\) to \(x = 2\pi\):
To create the graph from \(x = 0\) to \(x = 2\pi\):
- Start with the amplitude. Since our function is \(3 \cos(2x)\), the amplitude is 3, showing the graph oscillates between -3 and 3.
- Identify the period. With \(\cos(2x)\), the period is \(\pi\). Thus, one cycle of the wave occurs from \(x = 0\) to \(x = \pi\) and repeats from \(x = \pi\) to \(x = 2\pi\).
- Mark key points using intervals. Divide the period \(\pi\) into quarters to identify critical points where the cosine changes value (such as from 3, to 0, to -3, etc.).
- Plot these key points: \(x = 0\, (3)\), \(x = \frac{\pi}{4}\, (0)\), \(x = \frac{\pi}{2}\, (-3)\), \(x = \frac{3\pi}{4}\, (0)\), \(x = \pi\, (3)\), and continue for the next cycle.
