Verified Answer :
- Recognize that the angle \(\theta = -\frac{17\pi}{3}\) is a negative rotation.
- Calculate the coterminal angle by adding multiples of \(2\pi\) until the result falls in the range \([0, 2\pi)\), giving \(\theta' = -\frac{17\pi}{3} + 6\pi = \frac{\pi}{3}\).
- Identify that \(\frac{\pi}{3}\) lies in the first quadrant.
- Final answer: \(\boxed{\frac{\pi}{3}}\).
Explanation
- Understand standard position and coterminal angles.
To determine the position of the angle \(\theta = -\frac{17\pi}{3}\) radians in standard position, we first recall that an angle is in standard position when its vertex is at the origin and its initial side lies along the positive \(x\)-axis. A negative angle value signifies that the rotation occurs in the clockwise direction from the initial side. Because a full revolution of a circle corresponds to \(2\pi\) radians, we can find an equivalent (coterminal) angle within the standard interval \([0, 2\pi)\) by adding appropriate multiples of \(2\pi\) to our given angle.
- Calculate the coterminal angle.
We seek an integer \(k\) such that \(\theta' = \theta + 2k\pi\) satisfies \(0 \le \theta' < 2\pi\).
Starting with \(\theta = -\frac{17\pi}{3}\):
\[-\frac{17\pi}{3} + 2k\pi \ge 0\]
\[-\frac{17\pi}{3} + \frac{6k\pi}{3} \ge 0\]
\[-17 + 6k \ge 0\]
\[6k \ge 17\]
\[k \ge \frac{17}{6} \approx 2.83\]
Choosing the smallest integer \(k=3\), we find the coterminal angle \(\theta'\):
\[\theta' = -\frac{17\pi}{3} + 3(2\pi)\]
\[\theta' = -\frac{17\pi}{3} + 6\pi\]
\[\theta' = -\frac{17\pi}{3} + \frac{18\pi}{3} = \frac{\pi}{3}\]
Thus, the angle \(\theta' = \frac{\pi}{3}\) is coterminal with the original angle \(\theta\).
- Identify the quadrant and conclude.
Having determined the coterminal angle \(\theta' = \frac{\pi}{3}\), we now identify its position. The angle \(\frac{\pi}{3}\) radians is equivalent to \(60^\circ\). Since \(0 < \frac{\pi}{3} < \frac{\pi}{2}\), the terminal side of the angle lies in the first quadrant of the Cartesian coordinate system. Therefore, the angle \(-\frac{17\pi}{3}\) shares the same terminal side as \(\frac{\pi}{3}\), placing it in the first quadrant. The final answer is \(\boxed{\frac{\pi}{3}}\).
Examples
Understanding coterminal angles is essential in fields like navigation and engineering. For example, if a mechanical gear rotates \(-\frac{17\pi}{3}\) radians, you need to know its final orientation to align it correctly with another component. By finding the coterminal angle \(\frac{\pi}{3}\), you realize the gear has effectively rotated \(60^\circ\) from its reference point, which is critical for precise mechanical assembly and calibration.