To find the period of the sine function that has a maximum at \((\frac{\pi}{14}, 1)\) and a minimum at \((\frac{3\pi}{14}, -1)\), we start by noting that the sine function has specific characteristics:
- The maximum occurs at \(\frac{\pi}{14}\) and has a value of 1.
- The minimum occurs at \(\frac{3\pi}{14}\) and has a value of -1.
The distance between a maximum and the subsequent minimum in a sine function typically corresponds to a quarter of the period. Thus, we can calculate the distance between these two points:
\[ \text{Distance} = \frac{3\pi}{14} - \frac{\pi}{14} = \frac{3\pi - \pi}{14} = \frac{2\pi}{14} = \frac{\pi}{7}. \]
Since this distance represents a quarter of the period \(T\) of the sine function, we can set up the relationship:
\[ \frac{T}{4} = \frac{\pi}{7}. \]
To find the full period \(T\), we multiply both sides by 4:
\[ T = 4 \cdot \frac{\pi}{7} = \frac{4\pi}{7}. \]
Thus, the period of the sine function is
\[ \boxed{\frac{4\pi}{7}}. \]
1 answer