Which of the following shows a $-\frac{17 \pi}{3}$ radian angle in standard position?

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01-11-2026
Mathematics

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Which of the following shows a $-\frac{17 \pi}{3}$ radian angle in standard position?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T17:16:25-05:00

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.