Linear velocity is defined as distance over a period of time. For instance if a person ran 1 mile or approximately 1600 meters in 7 minutes, the they would have covered about 230 meters per minute. Let's assume this person was running around a track. We could also measure their speed from a central angle and represent their speed as the amount of degrees (or radians) they ran around per unit time as well. This is considered angular speed. A perfect example of both are the hands on a clock. There is a relationship between arc length which we designate s, the radius r, and the angle  (theta). The relationship is . So the length of the arc (s) is equal to the radius of the circle the arc is on and what section of the pie it covers ( this is akin to how many degrees our track runner ran "through" or around) . Thus, if we wanted the linear speed around a circle we could say  or . Key measurements that you will need to know are how many degrees in a circle of which there are 360 or . Before you follow the step by step solution below, go back and see if you can use this new information to arrive at the correct answer.

The actual answer is .

To see why, note that the second hand spins around a total of 360 degrees or .

.

And how long does it take for the hand to go around? Linear speed of the clock second hand is 

  (rounded answer).

Linear speed is equal to the arc length traversed divided by the time. We use s to denote arc length.

Write the conversion factor between radians and degrees.

Cancel the radians unit by using dimensional analysis.

Write the conversion factor of radians and degrees.

Substitute the degree measure into .

Firstly, since the point  is in the 3rd quadrant, it'll be between  and . If we take  to be the horizontal, we can form a triangle with base and leg of values  and . Solving for the angle in the 3rd quadrant given by , 

Since this angle is made by assuming  to be the horizontal, the total angle measure  is going to be:

In order to solve for the degree measure from radians, replace the  radians with 180 degrees.  

The nearest degree is .

A triangle has three angles that will add up to  degrees.

Convert the radians angle to degrees by substituting  for every .

The third angle is 60 degrees.

Let the second angle be .  The first angle three times the value of the second angle is .  Set up an equation that sums the three angles to .

Solve for .

Substitute  for the first angle and second angle.

The second angle is:  

The first angle is:  

The three angles are:  

The second highest angle is:  

The coterminal angles can be positive or negative.  To find the coterminal angles, simply add or subtract 360 degrees as many times as needed from the reference angle.

All of these angles are coterminal angles.

To find the coterminal angle of an angle, simply add or subtract  radians, or 360 degrees as many times as needed.

These are all coterminal angles to  radians.

Out of the given answers,  is the only possible answer.

In order to find a coterminal angle, simply add or subtract  radians to the given angle as many times as possible.

The possible angles after adding increments of  radians are:

The possible angles after subtracting decrements of  radians are:

Out of the given possibilities, only  is a valid answer.