Note: Since my blog editor can't really handle superscripts, I'll denote the inverse sine by "arcsine" and similarly for the other inverse trig functions.
First off, let me make a few remarks concerning how a slide rule handles the various trig functions. These are general observations, for in fact there is some variation on more advanced slide rules.
- The slide rule assumes degree measurement by default, although I'll show you how to work in radians as well.
- The format is decimal degrees, not degrees and minutes.
- The S scale handles sine, cosine, cosecant and secant for "larger" angles, i.e., between 5.74° and 90°.
- The T scale handles tangent and cotangent for "larger" angles, i.e., angles between 5.71° and 45°.
- The ST scale services all six of the functions just mentioned for "smaller" angles, i.e., angles less than 5.73°.
You might be thinking that those are pretty confining limits on the arguments. However, they're easily overcome by remembering:
- how to work with reference angles
- even and odd function properties
- the cofunction identities
Let's dig into the sine and arcsine functions.
Sine of an Angle Measured in Degrees
Our first example is a snap, and it doesn't even require any slide movement.
Example 1.1: Find sin(22.5°).
All of the action takes place on the C and S scales, but I'm still going to recommend you close the slide rule at the start (aligning C and D) so that the eyes don't get confused.
- Set the cursor at 22.5° on the S scale.
- Read the result under the hairline on the C scale: 0.383.
 |
| Cursor to S:22.5°, result at C:0.383 |
Your first thought no doubt is, "How do I know where to put the decimal point?" Good question, so let's look more closely at the situation.
On a slide rule, the arguments appearing on the S scale span 5.73° to 90°. These correspond to sine values of 0.1 to 1.0. We'll see how to deal with smaller numbers in the next problem.
Example 1.2: Find sin(1.8°).
Here we have a number less than 5.73°, so we switch scales and go with ST instead.
- Set the cursor to 1.8° on the ST scale.
- Read the result under the hairline on the C scale: 0.0314
 |
| Cursor to 1.8°, result at C:0.0314 |
Again, I hear you ask about the decimal point. For arguments from 0.573° to 5.73°, the value of sine ranges from 0.01 to 0.1. But let's try another, to really make the pattern stand out.
Example 1.3: Find sin(0.3°).
This is also less than 5.73°, so again we'll use the ST scale.
- Set the cursor to 0.3° (3°) on the ST scale.
- Read the result under the hairline on C: 0.00524.
 |
| Cursor to 0.3°, result at C:0.00524 |
Once more you ask about the decimal point. And in a somewhat repetitive fashion, I'll respond that for arguments between 0.0573° and 0.573° the sine's value spans 0.01 to 0.1.
Let's pull this all together. The ST scale is in fact a copy of the C and D scales, just shifted over by 5.73 and labeled a little differently. (I urge you to read more about this in Degrees, Radians and Arc Lengths). Thus, it can be used at any order of magnitude. If we consider the three problems we've just seen in reverse order, then:
- Angles from 0.0573° to 0.573° yield sines from 0.001 to 0.01
- Angles from 0.573° to 5.73° yield sine values from 0.01 to 0.1
- Angles from 5.73° to 90° yield sine values from 0.1 to 1.0.
Make sense now?
One final thing before we move on. You will have noticed that for angles less than 5.73° we're using the ST scale and that this is actually just an arc length measurement. No fancy trig, nothing transcendental, just a radian measurement as a decent approximation. The reason is found in a Calculus I course: the limit of sin(θ)/θ as θ goes to 0 (with θ in radians) is 1. In short, for small angles, the radian measure of an angle closely approximates the sine of the that angle.
Now back to our regularly scheduled programming.
Arcsine in Degrees
Pretty clearly we can work this in reverse to find the inverse sine of a number. Let's try some problems. By the way, we're still in degree mode, although strictly speaking in most undergraduate mathematics classes, the arcsine is defined as a real-valued function of a real variable.
Exercise 2.1: Find arcsin(0.2), in degrees.
For this one, we'll simple crank backwards from C to S.
- Set the cursor at 0.2 (2.0) on the C scale.
- Read the result under the hairline on S: 11.5°.
 |
| Cursor to C:0.2, result at S:11.5° |
Remembering what we learned in the previous section, I'm assuming you had no trouble interpreting the result as 11.5° and not 1.15°. Just think about that 0.2 and how it lies between 0.1 and 1.0, so the corresponding angle must lie between 5.73° and 90°.
Let's try another.
Example 2.2: Find arcsin(0.02), in degrees.
- Set the cursor at 0.02 (2.0) on the C scale.
- Read the result under the hairline on the ST scale: 1.15.
 |
| Cursor to 0.02, result at ST:1.15 |
Notice anything? It's the exact same setup as the previous problem, only now we read off the final value on ST. You see, the argument 0.02 lies between 0.01 and 0.1, so the answer appearing on the ST scale should be interpreted between 0.573 and 5.73. Refer to the bulleted items in Example 1.3, above for review.
Sine of an Angle Measured in Radians
If truth be told, the slide rule really doesn't care much for trig arguments in radians. Be we can coax it along with just a slight bit of dither.
Example 3.1: Find sin(0.5).
What we'll do here is simply convert radians to degrees at the outset (using the ST scale; see Degrees, Radians and Arc Length for an explanation), then proceed to the S scale to find the sine of that angle.
- Set the cursor at 0.5 (5.0) on the C scale
- Read 28.6 (2.86) under the hairline on the ST scale. This is the degree measurement of 0.5 radians.
- Leaving the cursor in place, align 28.6° on the S scale with the hairline.
- Read the result under the hairline on the C scale: 0.479.
 |
| Cursor at C:0.5, read ST:28.6°, leave cursor untouched |
 |
| Align S:28.6° with hairline, result at C:0.479 |
This is our first example of a number-transfer; we had to read that 28.6 on the ST scale, then find it on the S scale. Not particularly elegant, but at least all it took was moving the slide, leaving the cursor untouched. But, rather importantly, we got the slide rule to work a trig problem in radians.
As for setting the order of magnitude at the end, I'll refer you back to the explanation above. Let's try another.
Example 3.2: Find sin(1.2).
- Set the cursor at 1.2 on the C scale
- Read 69° under the hairline on the ST scale. This is the degree measurement of 1.2 radians.
- Leaving the cursor in place, align 69° on the S scale with the hairline.
- Read the result under the hairline on the C scale: 0.93.
 |
| Cursor at C:1.2, read ST:69°, leave cursor untouched |
 |
| Align S:69° with hairline, result at C:0.93 |
Kind of a tough one to nail accurately, isn't it! For one thing, the divisions on the ST scale are rather spread out in step 2. Then they bunch up like crazy on the S scale in step 3. However, you should be able to get at least one or two places of accuracy if you've got your reading spectacles on.
Give the next one a try, and do it in less than five seconds...
Example 3.3: Find sin(0.05).
Did I catch you? There's nothing to do, so don't even pick up the slide rule. We know for angles this small (again, see the ranges I mentioned earlier) the sine is within epsilon of the angle itself. So, the answer here is simply 0.05.
One final thought: when computing sines in radians, you'll want the argument to be less than or equal to 1.57, which corresponds to 90°.
Arcsine in Radians
Thinking in reverse, we should be able to take what we just learned and work backward from a sine value to the angle in radians it corresponds to. In a nutshell, find the degree version first, then convert to radians.
Example 4.1: Find arcsin(0.65), in radians.
- Set the cursor at 0.65 (6.5) on the C scale.
- Read 40.5° under the hairline on the S scale.
- Leaving the cursor in place, align 40.5° of the ST scale with the hairline.
- Read the result under the hairline on the C scale: 0.71.
 |
| Cursor to C:6.5, read S:40.5°, leave cursor untouched |
 |
| Align ST:40.5° with hairline, result at C:0.71 |
Example 4.2: Find arcsin(0.02), in radians.
Did I fool you again? Because of the small argument, this one is done before it even begins. The answer is 0.02 radians.
That wraps up sine and arcsine, in both degrees and radians. Stop back next time when we see how to wrangle its sister, the cosine.
Next installment: Cosine and Arccosine