The function \( y = \cos(2x) \) represents a cosine function with an angular frequency that is twice the normal frequency of the basic cosine function \( y = \cos(x) \).

Here are some important characteristics of this function:

1. **Amplitude**: The amplitude of \( \cos(2x) \) is 1, since there is no vertical scaling factor in front of the cosine function.
   
2. **Period**: The period of \( \cos(x) \) is \( 2\pi \), but for \( \cos(2x) \), the period is shortened. The period of \( \cos(kx) \) is given by \( \frac{2\pi}{|k|} \). Therefore, the period of \( \cos(2x) \) is:
   \[
   \text{Period} = \frac{2\pi}{2} = \pi
   \]
   This means that the function completes one full cycle over an interval of \( \pi \).

3. **Frequency**: The frequency is the reciprocal of the period. Since the period is \( \pi \), the frequency is \( \frac{1}{\pi} \).

4. **Phase Shift**: There is no phase shift, as there is no horizontal translation.

5. **Vertical Shift**: There is no vertical shift, since no constant is added or subtracted to the function.

Would you like more details on graphing or properties of this function?

Here are some related questions to deepen understanding:

1. How does changing the coefficient of \( x \) in \( \cos(kx) \) affect the graph?
2. What is the difference in the period between \( \cos(x) \) and \( \cos(3x) \)?
3. How would the function \( y = 2 \cos(2x) \) differ from \( y = \cos(2x) \)?
4. Can you explain how the derivative of \( y = \cos(2x) \) would look?
5. What happens to the graph of \( y = \cos(2x) \) if we add a phase shift of \( \frac{\pi}{4} \)?

**Tip**: The angular frequency \( 2x \) compresses the graph horizontally by a factor of 2, meaning the function oscillates more quickly.

Explore the properties of the cosine function y = cos(2x) with a detailed analysis of its amplitude, period, and frequency. Learn how angular frequency affects the graph and understand key formulas for solving trigonometric functions. Suitable for students in Grades 10-12.

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