Degrees, Radians and Arc Length

A typical course in trigonometry spends the first day or two on the subjects of degree-to-radian and radian-to-degree conversion, followed by some applications to arc length, area within a sector and angular velocity. None of these involves the trigonometric ratios, but the topics fit more closely with a trig class than other freshman mathematics offerings. So, that's where we'll start as we commence our journey through trigonometry on a slide rule.

Radians to Degrees


Let's begin with the problem of converting radian measure of an angle to degrees. On a typical slide rule, there are at least two ways to handle this. Recall that the conversion is suggested by π radians = 180°. Thus, when converting from radians to degrees, the constant 180°/π is the number we need, and this is about 57.3°.

Method #1: The first approach is really nothing more than a simple multiplication by the special constant just mentioned, carried out in the usual manner. Rather conveniently, some slide rules, like the Pickett 1006-ES I'm using throughout this blog, supply a gauge mark for the quantity 57.3° on the C and D scales. If that's not the case with yours, simply remember the number. Anyway, on my Pickett, it's denoted by the symbol R:



Exercise 1.1: Convert 3.9 radians to degrees.

We'll just use canonical multiplication here.
  1. Set the cursor at 3.9 on the D scale.
  2. Align the right index with the hairline.
  3. Move the cursor to R (57.3°) on the C scale.
  4. Read the result under the hairline on the D scale: 223°, after estimation to interpret the order of magnitude.
It looks like the following (which you can click to enlarge):

Cursor to D:3.9, right index to hairline, cursor to C:R, result at D:223°

Let's try it again, but at a different order of magnitude. 

Exercise 1.2: Convert 0.25 radians to degrees.

Commute the multiplication this time. You'll see why in just a moment.
  1. Set the cursor at R (57.3°) on the D scale.
  2. Align the right index with the hairline.
  3. Move the cursor to 0.25 (2.5) on the C scale.
  4. Read the result under the hairline on the D scale: 14.3°, after setting the decimal point.
In summary:

Cursor to D:R, right index to hairline, cursor to C:2.5, result at D:14.3°

Notice that by setting up the proportion with R first in step 1, you can then proceed to convert other angles without changing the position of the slide. For example, move the cursor to π on the C scale, then read 180° on the D scale at once. You can continue doing any number of radian-to-degree conversions this way. (You might need an index-swap, however, if you run off-scale to the left).

Method #2: You're going to love this method for converting radians to degrees, for there's no slide movement involved. It all hinges upon the ST scale, which is usually touted as existing for finding approximations to the sine or tangent of small angles. But in fact, the ST scale is also ideal for degree/radian conversion, as well as automating calculations involving arc length along a unit circle.

Let's investigate. Close the rule, then compare C to ST. (On other slide rules, you might be comparing D to ST, so adjust accordingly). Align the cursor with C:π and observe ST reads 180°. Then try C:6.28 (about 2π) and you'll see 360° on ST.

Get the picture? ST is in fact just a copy of the C scale, but shifted over by 5.73 logarithmic units. (Like most slide rule operations, the order of magnitude is irrelevant, so you can think of the shift as 0.573, 5.73, 57.3, etc.) In other words, the ST scale is preset to do multiplications (with respect to C) by 57.3° (i.e., 180°/π). Hey presto; automatic conversions from radians to degrees!

You might compare this to the CF scale that we met earlier. (See Method #3 in Multiplication by π and π/4). In that case, the shift is π units, setting things up for rapid multiplication by π. The point is, C, ST and CF are really all the same scale, but the latter two have been slid over to generate those special products: one by 180°/π, the other by π. Ingenuous but powerful!

Let's try it out on Exercise 1.1 from above.
  1. Close the rule (i.e., align C and D for ease on the eyes).
  2. Set the cursor at 3.9 on the C scale.
  3. Read the result under the hairline on the ST scale: 223°.
It looks like this:

Cursor to C:3.9, result under hairline at ST:223°

Of course, you need to mentally place the decimal point at the conclusion

Anyway, we've got a really slick approach here. For further practice, here's Exercise 1.2 from above using it:
  1. Close the rule (i.e., align C and D for ease on the eyes).
  2. Set the cursor at 0.25 (2.5) on the C scale.
  3. Read the result under the hairline on the ST scale: 14.3°.
On my Pickett 1006-ES I see:

Cursor to C:2.5, result under the hairline at ST:14.3°

As mentioned, seen in this light, we have a very direct method for going from a radian measure (on C) to the equivalent degree measure (on ST). And the technique is good for any order of magnitude, just by setting your decimal point properly at the end of the computation.

Now there's absolutely no reason why we can't work either of these two methods in the opposite direction, thus converting degree measure to radians. Let's check it out with Method #1 first. (I'll remind you again, on some slide rules you might be working with D and ST, so make any mental adjustment needed as we go along).

To convert a measurement from degrees to radians requires a multiplication by π/180°, or about 0.0174. That's not my favorite number to remember, but rather neatly, Pickett decided to include it as a gauge mark. The manufacturer indicates it by 1° on the ST scale:



This gauge mark aligns with 0.0174 (1.74) on the C scale, and if the rule is closed, with the D scale as well. Let's take it out for a test drive.

Exercise 2.1: Convert 75° to radians.

Using the reverse steps of Method #1 from above, we have:
  1. Close the rule (i.e., align C and D).
  2. Move the cursor to 1° on the ST scale.
  3. Align the right index with the hairline.
  4. Move the cursor to 75° (7.5) on the C scale.
  5. Read the result under the hairline on the D scale: 1.31 radians.
Pictorially:

Close rule, cursor to ST:1, right index to hairline, cursor to C:7.5, result at D:1.31

For practice, let's try it again but with a considerably smaller argument.

Exercise 2.2: Convert 0.33° to radians.
  1. Close the rule (i.e., align C and D).
  2. Move the cursor to 1° on the ST scale.
  3. Align the left index with the hairline.
  4. Move the cursor to 0.33° (3.3) on the C scale.
  5. Read the result under the hairline on the D scale: 0.00576 radians, after adjusting for the order of magnitude.
Here's how it appears on my Pickett 1006-ES:

Close rule, cursor to ST:1°, left index to hairline, cursor to C:3.3, result at D:0.00576

We can significantly cut down on the finagling by reversing Method #2 from above and using the ST scale. Let's see by trying Exercise 2.1 again.
  1. Close the rule to keep everything less confusing to the eyes.
  2. Align the cursor with 75° (7.5) on the ST scale.
  3. Read the result under the hairline on the C scale: 1.31 radians.
It's a cinch, as the following photo shows:

Cursor to ST:7.5, result under hairline at C:1.31

And for good measure, we'll run through Exercise 2.2 using this approach.
  1. Close the rule. I like to keep C and D aligned in general.
  2. Align the cursor with 0.33° (3.3) on the ST scale.
  3. Read the result under the hairline on the C scale: 0.00576, after adjusting for the order of magnitude.
And here is a photographic recap:

Cursor to ST:3.3, result under hairline at C:0.00576

Application to Arc Lengths


We know by definition that an angle of 1 radian subtends an arc of length 1 on the unit circle. Or if you prefer, think of that angle as 57.3°. Thus, the ST scale also provides a path for finding arc length when the angle is measured in degrees. Some rules like those from Aristo and Faber-Castell even label the ST scale as "arc."

Moreover, this can be extended to circles of any radius by means of proportionality. Hence, s = r θ, where s is the length of the arc, r is the radius of the circle in question and θ the measure of the subtending angle in radians. But now we know how to get the get the angle in radians from degrees and can really go to town. Let's see.

Exercise 3.1: Find the length of arc subtended by an angle of 50° on a circle of radius 30.
  1. Close the rule. 
  2. Set the cursor on 50° (5.0) on the ST scale.
  3. Align the right index with the hairline.
  4. Move the cursor to 30 (3.0) on the C scale.
  5. Read the result under the hairline on the D scale: 26.2, after adjusting the decimal point.
Now that was easy!

Close rule, cursor to ST:5.0, right index to hairline, cursor to C:3.0, result at D:26.2

Area within a Sector


Still in our "first day of trigonometry class" mode, let's see how to find the area enclosed within a sector using the slide rule. Recall the formula we need is: A = ½ θ r². This, too, can be proven by setting up a proportion involving the area within a circle. Here's the problem.

Exercise 4.1: Find the area within the sector of a circle of radius 18, subtended by an angle of 50°.
  1. Close the rule.
  2. Set the cursor at 50° (5.0) on the ST scale.
  3. Align 18 (1.8) of the CI scale with the hairline.
  4. Move the cursor to 18 (1.8) on the C scale.
  5. Read the result under the hairline on the D scale: 283 squared units.
This is actually an example of a chained operation, a topic I haven't officially gotten to yet, but it won't hurt for you to see it in action. Also, on my Pickett 1006-ES, the ST and CI scales are on opposite sides, so I had to do a duplex-flip at the start of step 2. For these reasons, it'll require three photos to show all the action.

Rule closed, cursor to ST:5.0, duplex-flip
CI:1.8 to hairline
Cursor to C:1.8, result under hairline at D:283

In step 2, we multiply once by 18 via division with the reciprocal on CI. Then in step 3, we multiply by 18 again, this time by moving directly along the C scale. Thus, we factored in our r² as required, but in two steps. I couldn't see any way clear to do it in one step on the A or B scales. Can you?

Wrap-Up: This is a toss-up, I think. Using ST as a fixed scaling factor (against C) for degree/radian conversion is certainly sweet. However, having the gauge marks for 1 radian and 1° makes these quite useful in chained operations. I vote we become proficient with either method.

One final note. I see that Pickett has also included the gauge marks ' and " on the ST scale, presumably for dealing with minutes and seconds. But I've got so many other things to write up now, I'll have to return to those guys later. Stay tuned.  

Next installment: Sine and Arcsine