Cosine of an Angle Measured in Degrees
As its name suggests, cosine is the co-function of sine, and so cos(θ) = sin(90° - θ). Of course, we already know how to handle sines on the slide rule, so this'll be a cinch. Just think in terms of the "complement of an angle." So, if cos(75.5°) is the goal, simply look for sin(14.5°).
On many slide rules, the complements are labeled on the S scale, often in red to remind you that they run backwards. On my favorite Pickett 1006-ES, there's no red, but there are some "<" signs next to the numbers to suggest reverse-reading.
As a reminder, on a slide rule, arguments for the sine span 5.74° to 90°, so the cosines (reading right to left) must cover 0° to 84.26°. The values of both functions range between 0.1 and 1.0.
Also recall that the sine of small angles (0° to 5.74°) is accommodated by the ST scale, with values running from 0.0 to 0.1. Therefore, in the case of cosine, the so-called "small" numbers are actually 84.26° to 90°.
Reminder: The magic number on the ST scale is actually 5.73°, but we'll use it as an approximation to 5.74° (as well as the tangent's 5.71°).
With that, we're ready to get to the problem.
Exercise 1.1: Find cos(75.5°).
This is easy; just compute sin(14.5°), i.e, work with the complement.
- Set the cursor at 14.5° on the S scale.
- Read the result under the hairline on C: 0.250.
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| Cursor to S:14.5°, result at C:0.250 |
Setting the decimal point at the end is trivial because we know values of cosine between 0° and 84.27° lie in the range 0.1 to 1.
Exercise 1.2: Find cos(86.2°).
The complement of this angle is 3.8°, which we recognize as lying in the "small angle" range. So we'll switch over to the ST scale for this problem.
- Set the cursor at 3.8° on the ST scale.
- Read the result under the hairline on C: 0.0663.
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| Cursor to ST:3.8, result at C:0.0663 |
If it isn't clear, we know that the resulting values must lie between 0.0 and 0.1 when using the ST scale to compute sines and cosines, so the decimal point is set accordingly as 0.063 (not 0.63).
Arccosine in Degrees
It's pretty simple to work backwards and compute arccosines as well when calculating in degree mode. Remember, we're asking "what is the angle (in degrees) whose cosine is such-and-such." (As a side note, we're thinking as an engineer might here; a mathematician would probably want the result to be in radians).
Exercise 2.1: Find arccos(0.57), in degrees.
In a capsule, we'll start with C, work in reverse to S (which will give the arcsine), then form the complement to arrive at the arccosine.
- Set the cursor at 0.57 (5.7) on C.
- Read the number under the hairline on S: 34.8°.
- Form the complement: 55.2°.
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| Cursor at C:0.57, complement at S:34.8°, result is 55.2° |
Naturally, we used the S scale here, since the argument 0.57 lies between 0.1 and 1.0. And, it should be clear that step 2 actually found the arcsine, which we then carried on from to determine the arccosine.
Example 2.2: Find arccos(0.05), in degrees.
At once we recognize that the argument is quite small, so we'll be using the ST scale this time.
- Set the cursor at 0.05 (5.0) on the C scale.
- Read the number under the hairline on the ST scale: 2.87°.
- Form the complement: 87.13°.
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| Cursor at 0.05, complement at 2.87°, result is 87.13° |
Cosine of an Angle Measured in Radians
In the next problem, we'll convert the argument from radians to degrees at the outset, form the complement of it, then transfer that number to the S scale, reading the final result on C. It sounds worse than it is. Even though it requires a number-transfer, the cursor never moves. Here goes nothing...
- Set the cursor at 0.7 (7.0) on the C scale.
- Read the number under the hairline on the ST scale: 40°. This is the radian measure of the argument.
- Mentally compute the complement: 50°.
- Leave cursor untouched, but slide 50° on the S scale under the hairline.
- Read the final result under the hairline on the C scale: 0.765.
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| Cursor at C:0.7, read under hairline ST:40°, form complement: 50° |
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| Don't touch cursor, align S:50° with hairline, result at C:0.765 |
Like I say, it really ain't all that bad; just run through a couple more examples and it becomes second nature. The important thing is, we now know how to make the slide rule compute these beasts in radians, a task that seemed dicey at first!
Arccosine in Radians
Ponder Example 3.1 and imagine you're working from the conclusion back to the start and you'll understand how to find the arccosine in radians. The basic plan makes perfect sense: find the arccosine of the argument in degrees, then convert that to radians at the end. I'll remind you that such conversions were explicated completely in Degrees, Radians and Arc Length.
Example 4.1: Find arccos(0.33), in radians.
- Set the cursor at 0.33 (3.3) on the C scale.
- Read arcsine in degrees under the hairline on S: 19.3°.
- Mentally convert to the complement, 70.7°, which is the arccosine in degrees.
- Leave the cursor untouched, but move the slide to align 70.7° (7.07°) on the ST scale with the hairline.
- Read the result under the hairline on the C scale: 1.23.
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| Cursor at C:0.33, read S:19.3°, form complement 70.7° |
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| Leave cursor untouched, align ST:70.7° with hairline, result at C:1.23 |
Once again, while this may read gruesomely, the process is actually quite straightforward and you can become fleet with it in a few moments of practice.
With that, we're ready to move on to the tangent and arctangent, which I'll write up in the next day or so.
Next installment: Tangent and Arctangent