Set 8 Problem number 1

• Important Note on Notation

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

How many degrees are in each of the following angles:

• 1 radian
• `pi radians
• `pi /2 radians and
• `pi /6 radians?

Solution

A complete circle is 360 degrees. A complete circle is 2 `pi radians.

So 360 degrees must equal 2 `pi radians.

• 1 radian must therefore equal 360/(2 `pi ) degrees, or 57.29 degrees.

`pi / 2 radians is 1/4 of 2 `pi radians, or 1/4 of a circle.

• This is clearly equivalent to 90 degrees.

`pi /6 radians is 1/12 of 2 `pi radians, or 1/12 of a circle.

• This is equivalent to 1/12 of 360 degrees, or 30 degrees.

Generalized Solution

In general if an angle is `theta radians, then since a radian is 360 / (2 `pi) degrees = 180 / `pi degrees, we have an angle of `theta * (180 / `pi deg) = 180 `theta / `pi degrees.

Explanation in terms of Figure(s), Extension

The figure below shows a circle with a 1-radian sector.

• One radian is the angle such that the sector forms an equilateral 'rounded triangle', consisting of two radial lines from the center to the circle and the arc of the circle between these two lines.
• The distance along the arc is equal to the lengths of the two radial lines, which is equal to the radius of the circle.

Since the circumference of the circle is 2 `pi r, where r is the radius of the circle, we can fit 2 `pi arcs each of length r around the circle.

• The angle corresponding to the complete circle is therefore 2 `pi radians, which is therefore equal to 360 degrees.

We reason out equivalent angles as follows:

• Since 2 `pi radians = 360 deg, a radian is 360 / (2 `pi) degrees.
• Since 1/4 of a circle is 90 deg, 90 deg is 1/4 of 2 `pi rad, or `pi / 2 rad.
• Since 1/12 of a circle is 30 deg, 30 deg is 1/12 of 2 `pi rad, or `pi / 6 rad.
• Since 1/8 of a circle is 45 deg, 450 deg is 1/8 of 2 `pi rad, or `pi / 4 rad.

Figure(s)

Important Note on Notation

The most common Greek symbols used in describing rotational motion, and some of the equations using these symbols, are summarized on the two tables below.  You should make careful note of these symbols for reference throughout this problem set.