Math Problem Statement

What is the period of the function f(x) = cos(8x)?

Solution

To find the period of the function f(x)=cos(8x)f(x) = \cos(8x), we use the formula for the period of cosine functions:

Period=2πb\text{Period} = \frac{2\pi}{|b|}

Here, bb is the coefficient of xx in the argument of the cosine function. In this case, b=8b = 8.

Period=2π8=π4\text{Period} = \frac{2\pi}{8} = \frac{\pi}{4}

Thus, the period of f(x)=cos(8x)f(x) = \cos(8x) is:

π4\boxed{\frac{\pi}{4}}

Let me know if you'd like further clarification or have any related questions! Here are some additional questions for exploration:

  1. What is the amplitude of f(x)=cos(8x)f(x) = \cos(8x)?
  2. How would the period change if f(x)=cos(kx)f(x) = \cos(kx) where k=12k = 12?
  3. What is the phase shift of f(x)=cos(8x+π/3)f(x) = \cos(8x + \pi/3)?
  4. How does the graph of f(x)=cos(8x)f(x) = \cos(8x) compare to f(x)=cos(x)f(x) = \cos(x)?
  5. What is the frequency of f(x)=cos(8x)f(x) = \cos(8x)?

Tip: For trigonometric functions, always identify the amplitude, period, and phase shift to fully describe the function.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Periodic Functions

Formulas

Period of cosine: Period = 2π / |b|, where b is the coefficient of x

Theorems

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Suitable Grade Level

Grades 10-12