What is the period of the cosine function, $$ y = \cos(2x) $$?

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UpStudy AI · Accepted Answer

The **period** of a trigonometric function is the length of one complete cycle of the function. For the cosine function, determining the period involves understanding how the function behaves with respect to its frequency multiplier.

### General Form:
The general form of a cosine function is:
$$ y = \cos(kx) $$
where:
- $$ k $$ is the **frequency multiplier**.

### Period Formula:
The period ($$ T $$) of the cosine function is given by:
$$ T = \frac{2\pi}{|k|} $$

### Applying to $$ y = \cos(2x) $$:
In the given function:
$$ y = \cos(2x) $$
- Here, $$ k = 2 $$.

Plugging this value into the period formula:
$$ T = \frac{2\pi}{2} = \pi $$

### Conclusion:
The period of the cosine function $$ y = \cos(2x) $$ is **$$ \pi $$**.

**Visualization:**
- While the standard cosine function $$ y = \cos(x) $$ completes one cycle from $$ 0 $$ to $$ 2\pi $$, the function $$ y = \cos(2x) $$ completes a cycle in half that distance, from $$ 0 $$ to $$ \pi $$.

**Final Answer:**
The period of $$ y = \cos(2x) $$ is $$ \pi $$.
The period of $$ y = \cos(2x) $$ is $$ \pi $$.

What is the period of the cosine function, ?

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