cos 4 pi
Understand the Problem
The question asks for the value of the cosine function at 4 pi radians. This relates to the periodic properties of the cosine function on the unit circle, where multiple angles correspond to the same point.
Answer
The value of $\cos(4\pi)$ is $1$.
Answer for screen readers
The value of the cosine function at $4\pi$ radians is $1$.
Steps to Solve
- Identify the angle in standard position
To find the value of the cosine function at $4\pi$ radians, we first need to determine its equivalent angle within the range of $[0, 2\pi]$. The cosine function has a periodicity of $2\pi$, meaning $cos(x) = cos(x + 2\pi k)$ for any integer $k$.
- Calculate the equivalent angle
Since $4\pi$ is greater than $2\pi$, we can subtract $2\pi$ from $4\pi$ to find its equivalent angle within the first full rotation (or $[0, 2\pi]$):
$$ 4\pi - 2\pi = 2\pi $$
This tells us that $4\pi$ radians is equivalent to $2\pi$ radians.
- Find the cosine of the equivalent angle
Now, we need to find the cosine of $2\pi$. The point corresponding to $2\pi$ on the unit circle is $(1, 0)$.
The cosine value is the x-coordinate of the point on the unit circle:
$$ \cos(2\pi) = 1 $$
The value of the cosine function at $4\pi$ radians is $1$.
More Information
The cosine function cycles every $2\pi$ radians, which is why we can subtract $2\pi$ to find an equivalent angle. This property makes it easier to evaluate trigonometric functions at larger angles.
Tips
- A common mistake is attempting to calculate $\cos(4\pi)$ without recognizing its periodic nature. Always simplify angles using the periodicity of trigonometric functions.
