Cosecant and Arccosecant

Now we'll turn our attention to the so-called "reciprocal" trig functions, beginning with cosecant and its inverse. As usual, we'll learn how to use these in either degrees or radians.

In a nutshell: csc(θ) = 1/sin(θ), and so we'll clearly be using the CI scale for inverses. Since I've covered sine in such excruciating detail earlier, we can really breeze through cosecant. I'll refer you back to Sine and Arcsine if you need some review. About all that's new to be considered here is:
  1. When 5.74° < θ < 90°, the cosecant lies between 1 and 10 (10 and 1 if you want to get persnickety about the order). Use the S scale.
  2. When θ < 5.74°, the cosecant is greater than 10). Use the ST scale.
In either case, grab the desired result from CI.

With that, let's begin!

Cosecant of an Angle Measured in Degrees


Exercise 1.1: Find csc(37.2°).

Do this exactly like you would for sine, except fetch the result from CI rather than C.
  1. Set the cursor at 37.2° on the S scale.
  2. Read the result under the cursor on the CI scale: 1.65.
Note: On my Pickett 1006-ES slide rule, shown here, it's faster to use DI (with the rule closed) than CI; this obviates the duplex-flip otherwise required. But go ahead and use whichever is more convenient for you.

With that in mind, when I worked the problem, it looked like this (click the photo to enlarge):

Cursor to S:37.2°, result at DI:1.65

Exercise 1.2: Find csc(1.5°).

Since this is a small angle (less than 5.74°) we know to use ST here.
  1. Set the cursor at 1.5° on the ST scale.
  2. Read the result under the hairline on CI: 38.2.
I'll use the DI scale instead again on my Pickett:

Cursor to ST:1.5°, result at DI:38.2

Arccosecant in Degrees


The main thing to remember here is to use the S scale when the argument is less than 10, and the ST scale otherwise.

Exercise 2.1: Find arccsc(1.05), in degrees.
  1. Set the cursor at 1.05 on the CI scale.
  2. Read the result under the hairline on the S scale: 72°.
Here I am using DI again, equivalent to CI when the rule is closed, for convenience:

Cursor to DI:1.05, result at S:72°

Exercise 2.2: Find arccsc(10.5), in degrees.

You'll note that the setup is identical to the previous problem, but now we snag the result from the ST scale. That the argument is larger than 10 alerts us to this.
  1. Set the cursor to 10.5 (1.05) on the CI scale.
  2. Read the result under the hairline on the ST scale: 5.47°.
Graphically, we have:


Cursor to DI:10.5, result at ST:5.47°

Cosecant of an Angle Measured in Radians


If you're feeling queasy about angle conversions, take a moment to review Degrees, Radians and Arc Length before proceeding.

Exercise 3.1: Find csc(0.66).

Clearly the way to begin is by converting that 0.66 radians to degrees, and then carrying on.
  1. Set the cursor at 0.66 (6.6) on the C scale.
  2. Read the number under the hairline of the ST scale: 37.9°. Leave the cursor untouched.
  3. Move the slide to align 37.9° of the S scale with the hairline.
  4. Read the result under the hairline on the CI scale: 1.63.
In this case I had no option on my Pickett but to use the CI scale, requiring a duplex-flip. Hence the three pictures here.

Cursor to C:0.66, read ST:37.9°, leave cursor untouched
Align S:37.9° with hairline
Duplex-flip, result at CI:1.63

Arccosecant in Radians


The usual protocol applies here. First find the arccosecant in degrees, then convert to radians.

Exercise 4.1: Find arccsc(2.5), in radians.

Since we're searching for an arccosecant here, we begin on the CI (or DI) scale, rather than the C scale.
  1. Set the cursor at 2.5 on the CI scale. We know that the result must lie between 5.74° and 90°.
  2. So, read the number under the hairline on S as: 23.6°.
  3. Leave the cursor untouched.
  4. Move the slide so that 23.6° of the ST scale lies under the hairline.
  5. Read the result under the hairline on C: 0.41.
This'll take two pictures:

Cursor at CI:2.5, read S:23.6°, leave cursor untouched

Align ST:23.6° with hairline, result at C:0.41

Exercise 4.2: Find arccsc(15.7), in radians.

Before you reach for the slide rule, pause a moment to reflect on that argument; 15.7 is a pretty large number. Refer back to my opening comments on the expected ranges.

A moment's reflection ought to convince you that we're in the "small angle" range. We've already seen that sin(θ) ≈ θ in that span. Recalling that csc(θ) = 1/sin(θ) then we also know csc(θ) ≈ 1/θ if θ (in radians) is tiny.

Therefore, to solve this problem, all you have to do is find the reciprocal of 15.7 and you're done: 0.064 radians. (Note carefully that 15.7 lies between 10 and 100, so we know where to place the decimal point in the final value).

 Exercise 4.3: Find arccsc(157), in radians.

 Again, we must be leading up to a small angle. But since its cosecant, 157, is greater than 100, the desired angle must be smaller yet. Hence, after finding the reciprocal of 157, we decide upon 0.0064. Compare with the previous problem.

We're really making hay now. Coming up next will be the secant and its inverse which behave in much the same way.

Next installment: Secant and Arcsecant