Tutorial: Hyperbolic Functions

Hyperbolic Functions

For regular Trigonometry read this

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Instructions
Examples

Introduction

Hyperbolic functions are analogous to Trigonometric functions. As the point [cos u, sin u] describes a circle as u varies, the point [cosh u, sinh u] describes the right half of a hyperbola. Aside from that, the similarities are more algebraic than geometric. There are comparatively fewer problems which require hyperbolic functions, and the problems are often more advanced, thus hyperbolic scales are far less likely to appear on a given slide rule. Those slide rules which included a set of hyperbolic scales billed themselves as "Vector" slide rules (That's the V in the V1). It's not clear in what sense vectors imply hyperbolic functions, but that was the term used.

Just like the trig scales, the hyperbolic scales can be found on both the stock or the slider and are used in conjunction with the C/CI or D/DI scales.

There is not a separate set of scales for degrees,minutes,and seconds. If hyperbolic angles can be considered angles at all, they are akin to radians, not degrees, and are defined by areas on a particular geometric construction. They are certainly not related to a convenient number of equal divisions of a circle. The hyperbolic scales also lack the double nature of the trig scales, there are no forward and reverse scales, all scales increase only to the right.

Offscale values

Hyperbolic angles aren't cyclic, and there's no way to cover the entire range. The scales cover the ranges from 0 to about 3, which is where a lot of the interesting values occur. After angles higher than 5, the functions have either settled settled down to a constant value or have started to become very large and accurately approximated by exponential functions. The gap between 3 and 5 is a limitation of hyperbolic scales on the slide rule. Sh3,Ch2,and Th2 scales covering this range could easily be constructed, but historically this has never happened. There just aren't enough uses for hyperbolic functions to justify 6 scales on a rule, plus the LL3_E and LL03_E scales can find a larger range of angles with a little more work, making use of the definitions:

(The rest can be derived using identities exactly like those in trigonometry: tanh = sinh/cosh, sech = 1/cosh etc.)

The point is, for large offscale values, you can read e^6 directly from 6 D to LL3_E ( 403.43 ), divide by 2 and get 201.7, which is both sinh 6 and cosh 6 to four places, since e^-6 on LL03_E contributes next to nothing in this case.

Likewise offscale Sh1 and Th values very close to zero can be calculated with this same method, ignoring the contribution from LL3_E .

Being non-cyclic, there are no shifting or cyclic identities with the hyperbolic functions, and sinh is not just cosh shifted over 90º. However like their circular counterparts, cosh is an even function and sinh,tanh are odd functions. That is:

sinh(-x)= -sinh(x)

cosh(-x) = cosh(x)

tanh(-x) = -tanh(x)

Thus, negative angles are no problem. Rather than memorizing identities for every function, it may be easier to keep in mind the graphs of these functions.

Instructions

We use a configuration with the hyperbolic scales on the slider, and the C,CI scales as their counterparts. If your hyperbolic scales are on the stock, use D,DI instead.

In all cases, we assume the angle x positive. If it is negative, the identities in the previous section tell you when to negate the answer.

y = sinh x

find x on Sh1 or Sh2. Set cursor
read y on C at cursor.
divide y by 10 if on Sh1

y = csch x

find x on Sh1 or Sh2. Set cursor
read y on CI at cursor.
divide y by 10 if on Sh2

y = arcsinh x

only for 0.1 <= x <= 10
set cursor at coefficient of x on C
if x is less than 1, use Sh1
otherwise use Sh2
answer at cursor on Sh1 or Sh2.

y = arccsch x

only for 0.1 <= x <= 10
set cursor at coefficient of x on CI
if x is less than 1, use Sh2
otherwise use Sh1
answer at cursor on Sh1 or Sh2.

y = cosh x

find x on Ch, set cursor
read y on C at cursor

y = sech x

find x on Ch, set cursor
read y on CI at cursor
divide by 10

y = arccosh x

only for 1 <= x <= 10
set cursor at x on C
answer at cursor on Ch.

y = arcsech x

only for 0.1 <= x <= 1
set cursor at coefficient of x on CI
answer at cursor on Ch.

y = tanh x

find x on Th, set cursor
read y on C at cursor
divide by 10.

y = coth x

find x on Th, set cursor
read y on CI at cursor

y = arctanh x

only for 0.1 <= x <= 1
set cursor at coefficient of x on C
answer at cursor on Ch.

y = arccoth x

only for 1 <= x <= 10
set cursor at x on CI
answer at cursor on Ch.

This list is a little simpler than the equivalent list in the trig tutorial. Only Sh has two scales, and none of them have scales running in both directions.

Examples

sinh 2: set cursor at 2 on Sh23; read answer 3.6269 at cursor on C
arcsinh -0.5: we'll work with 0.5 and negate the answer, as arcsinh is an "odd" function.; set cursor at 0.5 on C; 0.5 is less than 1, so read answer 0.4812 on Sh1; negate answer: -0.4812