Which of the following shows a $-\frac{5 \pi}{6}$ radian angle in standard position?

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01-11-2026
Mathematics

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Which of the following shows a $-\frac{5 \pi}{6}$ radian angle in standard position?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T17:07:07-05:00

The angle $\theta = -\frac{5\pi}{6}$ radians is converted to degrees: $\theta = -\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = -150^\circ$.
The negative sign indicates a clockwise rotation starting from the positive x-axis.
Rotating $150^\circ$ clockwise from the positive x-axis places the terminal side in the third quadrant, specifically $30^\circ$ below the negative x-axis.
The angle corresponds to a positive rotation of $210^\circ$, which is in the $\boxed{\text{third quadrant}}$.

Explanation

Convert radians to degrees.

The angle given is $-\frac{5\pi}{6}$ radians. In standard position, an angle's vertex is at the origin $(0,0)$, and its initial side is the positive x-axis. A positive measure indicates a counterclockwise rotation, while a negative measure indicates a clockwise rotation.

To convert $-\frac{5\pi}{6}$ radians to degrees, we multiply by $\frac{180^\circ}{\pi}$:
\[-\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = -5 \times 30^\circ = -150^\circ\]
This means the terminal side of the angle is rotated $150^\circ$ clockwise from the positive x-axis.

Locate the terminal side position.

The positive x-axis is at $0^\circ$. Rotating $150^\circ$ clockwise:

After $90^\circ$ of clockwise rotation, the terminal side aligns with the negative y-axis ($-90^\circ$).
Continuing another $60^\circ$ clockwise ($150^\circ - 90^\circ = 60^\circ$), the terminal side lands in the third quadrant.
Specifically, the negative x-axis is at $-180^\circ$. The angle $-150^\circ$ is $30^\circ$ above the negative x-axis (since $-150^\circ = -180^\circ + 30^\circ$).
Equivalently, in a positive sense, $-150^\circ$ is equal to $360^\circ - 150^\circ = 210^\circ$. An angle of $210^\circ$ is $30^\circ$ into the third quadrant past the negative x-axis.

Conclusion.

The angle $-\frac{5\pi}{6}$ radian in standard position:

Starts at the positive x-axis.
Rotates clockwise by $150^\circ$.
Terminates in the third quadrant, forming an angle of $30^\circ$ with the negative x-axis.

The final answer is that the angle lies in the third quadrant. $\boxed{\text{third quadrant}}$.

Examples

Understanding standard position is essential in fields like navigation and mechanical engineering. For instance, if a radar system tracks an object at an angle of $-\frac{5\pi}{6}$ radians, the operator knows the object is in the third quadrant relative to their north-south (y-axis) and east-west (x-axis) reference lines. Calculating this helps the operator identify the object's general direction quickly.