sin 4 pi
Understand the Problem
The question is asking to find the value of the sine function at the angle of 4 pi radians.
Answer
$0$
Answer for screen readers
The value of the sine function at the angle of $4\pi$ radians is $0$.
Steps to Solve
- Identify the sine function's periodicity
The sine function is periodic with a period of $2\pi$. This means that the values of the sine function repeat every $2\pi$ radians.
- Reduce the angle
We can reduce the angle $4\pi$ radians using the periodicity of the sine function:
$$ 4\pi \mod 2\pi = 0 $$
This calculation shows that $4\pi$ is equivalent to $0$ radians.
- Evaluate the sine at the reduced angle
Now, we need to find the sine of the reduced angle:
$$ \sin(0) = 0 $$
The value of the sine function at the angle of $4\pi$ radians is $0$.
More Information
The sine function, represented by $\sin(x)$, gives the y-coordinate of a point on the unit circle corresponding to the angle $x$. Since $4\pi$ radians corresponds to a complete rotation around the circle, it returns to the starting point where the sine value is $0$.
Tips
- Failing to recognize the periodicity of the sine function, which can lead to incorrect evaluations for larger angles.
- Not reducing the angle before calculating the sine, which complicates the problem unnecessarily.
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