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In a typical physics class, we change from degrees to radians and back several times.  When should you use radians vs. degrees?  I’ll help you decide.

You should use radians when you are looking at objects moving in circular paths or parts of circular path.  In particular, rotational motion equations are almost always expressed using radians.  The initial parameters of a problem might be in degrees, but you should convert these angles to radians before using them.

You should use degrees when you are measuring angles using a protractor, or describing a physical picture.  Most people have developed intuitive feel for the common angles.  This would be common in vector related problems, including speeds, projectiles, forces, and similar situations.

### Warning for Calculators

Your calculator has three angle-related options.

• DEG mode, which uses degrees in trig functions,
• GRAD mode, which breaks a circle into 400 pieces.

Make sure that your calculator is in the proper mode depending upon the topic you are studying.  Unless you are a surveyor, chances are that you will never use GRAD mode, except by mistake.

## Degrees

The circle was divided into 360 degrees in ancient times.  Several plausible reasons exist.

• The year has 365 days.
• The number 360 can be broken in to many factors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 45, 60, 90, 120, 180, 360.  This makes it easy to divide a circle into equal parts.  (This is similar to the reason why we divide a foot into 12 inches.)
• The number 360 is a natural part of the base 60 number system.

One of the results is that the size of a degree is relatively small. We can use a protractor and measure or draw the angles of a triangle relatively easily.  You can probably draw a triangle with angles of 30°, 60°, and 90° fairly easily.  You could also estimate the angle of a 45° incline fairly accurately.

Throughout the early weeks of a physics class, we use angles measured in degrees frequently.  Degrees show up with vectors and are used in directions, forces, accelerations, slopes, and other measurements.

What is the other choice in radians vs. degrees?

The radian is related to the diameter (and thus radius) of a circle.  In ancient times, people realized that the diameter D, and the circumference C of all circles had a common ratio, i.e.

$$\color{black}{ \frac{C}{d}=3.14159…}$$

Since this ratio was fixed, they gave it a special symbol, π.   It is known as an irrational number, because it can’t be written as a fraction with two integers.  (The ratio 22/7 comes about as close as you can get reasonably.)

You probably learned this relation using the diameter or the radius (2 = D) as…

$$\color{black}{C=\pi D \textrm{ or } C=2\pi r}$$

The radian is really unitless, because it is the ratio of two lengths.  However, when we are working with circles or parts of circles, we want to make sure we know we are talking about this ratio. Hence, we always add the word radians to show this ratio.

It is better to say that there are “2π radians” in a circle, rather than just “2π” in a circle.

The problem with radians is that an angle of one radian is fairly large.  In fact, you could estimate it at about 60 degrees, though its true value is about 57.32 degrees.  This would be like trying to measure the length of your fingers in yards.  You could do it, but each of your fingers would measure as some small fraction of a yard.

This is convenient in some cases though.  You probably remember using fractions like π/2 = 90°, π/3 = 60°, π/4 = 45°, and π/6 = 30° back in trigonometry class.

When we talk about rotational motion, radians become the preferred unit of measure for angles.  This ultimately stems from the description of the arc length, s, given by

$$\color{black}{ s = r \theta.}$$

Using this relationship, we can multiply the radius by the angle in radians to determine the arc length.  No conversions are necessary.

In particular, for uniform circular motion, the angular frequency, ω, is related to the period, T, through the relation

$$\color{black}{ \omega = \frac{2 \pi}{T},}$$

which tells us that if we know the time (period =T) it takes to move once around the circle (angle = 2π), we can determine the angular speed.  (For uniform circular motion, the speed is constant, so the angular speed and the angular frequency are the same.)

For parts of the circle, the distance traveled around the circle can be found using

$$\color{black}{ \omega = \frac{\theta}{t}=\frac{s}{rt}}$$

so that

$$\color{black}{ s=\omega r t.}$$

If we didn’t use radians, we would have to convert out of degrees for this to work.

### The Small Angle Approximation

We need to be measuring angles in radians to make use of the small angle approximation.  If you apply the approximation to compare degrees, it won’t work at all.

### Warning for Excel

Excel is programmed to assume that things are measured in radians when using any of the trig or inverse trig functions.  For example, if you take the sine of a number, Excel assumes that the number is an angle in radians.  Likewise, if you take the inverse sine (or arcsine) of a fraction, Excel will give you the result in radians.

In a later lesson, I’ll give you hints on how to spot these types of mistakes.  For now, I hope you are able to choose between radians vs. degrees.