At 7:00, what is the angular position in radians of the minute hand of a clock in standard position, measured from the positive $x$-axis (3 o'clock position) counterclockwise?

• A. $0$
• B. $\frac{7\pi }{6}$
• C. $\pi$
• D. $\frac{7\pi }{12}$

Answer: (A)

Verified step by step guidance

1. Understand that the minute hand completes a full circle (360 degrees or \(2\pi\) radians) every 60 minutes.
2. Calculate the angle moved by the minute hand per minute by dividing the full circle by 60: \(\frac{2\pi}{60} = \frac{\pi}{30}\) radians per minute.
3. At 7:00, the minute hand is exactly at the 12 o'clock position, which corresponds to 0 minutes past the hour.
4. Multiply the angle per minute by the number of minutes past the hour to find the angular position: \(0 \times \frac{\pi}{30} = 0\) radians.
5. Since the standard position measures angles counterclockwise from the positive x-axis (3 o'clock position), and the minute hand at 12 o'clock is 90 degrees (or \(\frac{\pi}{2}\) radians) from 3 o'clock, adjust the angle accordingly by subtracting \(\frac{\pi}{2}\) radians if needed to express the angle from the positive x-axis.