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Chapter 6: Problem 33

What is the period of the function \(\cos (2+x) ?\)

Short Answer

Expert verified
The period of the function \(\cos (2+x)\) is \(2\pi\).

Step by step solution

01

Determine the general form of the function

Rewrite the given function as \(\cos (kx)\), in this case, we have \(k=1\) and \(x=2+x\). So the function is \(\cos ((1)(2+x))\).
02

Identify the frequency of the function

The frequency of the given function is equal to the coefficient of the variable inside the cosine, which in our case is \(k=1\).
03

Find the period of the function

The period of a cosine function is given by the formula \(T=\frac{2\pi}{|k|}\). In our case, \(k=1\), so the period is \(T=\frac{2\pi}{|1|}=2\pi\). Therefore, the period of the function \(\cos (2+x)\) is \(2\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is one of the primary trigonometric functions. You often see it written as \(\cos(x)\). It's vital in mathematics, especially when dealing with angles and periodic phenomena.
When we talk about \(\cos(x)\), we often refer to how it behaves on the unit circle. Here's what you should know:
  • For any angle \(x\), \(\cos(x)\) gives the x-coordinate of where this angle intersects the unit circle.
  • The function is periodic, meaning it repeats its values in regular intervals. This is important when observing wave-like behaviors.
  • Cosine is an even function. This means that \(\cos(-x) = \cos(x)\), which is helpful in simplifying expressions.
Understanding these properties helps when working with transformations of the function, such as horizontal shifts and scaling. This forms a basis for tackling more complex trigonometric problems.
Period of a Function
The period of a function is the length of the interval over which it repeats itself. In the context of the cosine function, it's crucial for understanding the full cycle of its oscillation.
For the basic cosine function \(\cos(x)\), the period is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the cosine function's values repeat.
To find the period of a transformed cosine function \(\cos(kx)\), use the formula:
  • \(T = \frac{2\pi}{|k|}\)
  • Here, \(k\) is the coefficient of \(x\), and it dictates how stretched or compressed the function is.
So, in a function like \(\cos(2+x)\), rewriting it to \(\cos(1 \cdot (2+x))\) clarifies that \(k = 1\) and therefore, the period remains \(2\pi\). Recognizing the period is essential for graphing the function and predicting its behavior over intervals.
Frequency in Trigonometry
Frequency in trigonometry tells us how often a wave pattern repeats over a certain interval. In terms of trigonometric functions, frequency is closely related to the period.
In general:
  • The frequency \(f\) of a trigonometric function like \(\cos(kx)\) can be calculated using \(f = \frac{1}{T} = \frac{|k|}{2\pi}\).
  • Therefore, increasing \(k\) increases frequency, causing the wave to "happen" more quickly over a given length.
Putting it into context with \(\cos(2+x)\), since \(k = 1\), the frequency is \(\frac{1}{2\pi}\). This implies that one complete wave of the function occurs every \(2\pi\) units.
Understanding frequency helps in analyzing wave behavior and is especially useful in fields such as physics, engineering, and signal processing where waves play a key role.

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