9.17 Vector/Hyperbolic Calculations

Congratulations on making it to this section; and kudos if you make it to the end! Here we describe one of the more esoteric uses of a small collection of some of the mid-twentieth-century side rules. These so-called vector rules contain scales of the hyperbolic sine and hyperbolic tangent functions, typically noted by the symbols SH/TH, or Sh/Th. While the values of such functions can be useful in a variety of physics and engineering problems, these rules were particularly useful in electrical engineering where the computation of standard trigonometric and hyperbolic trigonometric operations on complex numbers or vectors are performed.

When we say complex numbers or complex vector calculations, we aren’t referring simply to “complicated” procedures; we are referring to operations on specific sets of numbers – vectors – that have, in our case, two components. A complex number or a vector in our present context can be thought of as an ordered-pair of two numbers, such as a set of coordinates for instance. On a map, moving to a point six miles away, 30 degrees North of East can be represented as a vector. On the map, one might draw an arrow from the original point to the new point; the arrow has a length corresponding to 6 miles, and is pointed at an angle of 30\(^\circ\) north of due-east – two numbers are required to define the vector. Getting to this point would also be equivalent to moving 5.2 miles East and 3 miles North – again, two numbers are involved.

A similar “picture” appears when discussing complex numbers. A complex number is one that involves the square root of -1. Any real number when “squared” will give a positive result: \(2^2 = 4\); \((-2)^2 = 4\). A number that when squared gives a negative result is called an imaginary number. It can be imagined as a real number times the square root of -1. The symbol \(i\) (for imaginary) is used to denote \(\sqrt{-1}\). For example \((2i)^2 = 4\times i^2 = -4\). The number -2\(i\) also gives -4 when squared. In engineering practice, the symbol \(j\) is often used rather than \(i\) but has the exact same meaning.

In general, a complex number is defined as one with both real and imaginary parts. They can be written in the form \(z = x + iy\). Often times complex numbers are used in engineering and scientific problems related to oscillating systems, which can have an amplitude and a relative phase of the oscillation for example. How they relate to a vector and to oscillations will become apparent below. First, though, we need to review the standard hyperbolic trigonometric functions and their relationship to the standard trig functions.

9.17.1 Hyperbolic Trigonometric Functions

We are familiar with the standard trigonometric functions sine, cosine and tangent which were discussed in Trigonometric Scales. Combinations of these functions are solutions to systems which oscillate in time with constant amplitudes. For example, consider a vector like the one in our figure above. It has an amplitude (6 miles, in the example), call it \(r\), and a phase angle (30 degrees in our example), call it \(\theta\). If one were to move along a circle of constant radius \(r\), then \(\theta\) would change and the horizontal and vertical coordinates (\(x\) and \(y\)) on the map would oscillate according to \[ x = r\cos\theta, ~~~~~~ y = r\sin\theta. \] However, for systems in which oscillations are being driven by external sources or damped due to friction or resistance, for instance, then the solutions can involve amplitudes that grow or reduce exponentially. These solutions involve the so-called hyperbolic trigonometric functions:

\[ \sinh(u) = \frac{e^{u}-e^{-u}}{2}; ~~~~~~~ \cosh(u) = \frac{e^{u}+e^{-u}}{2} ; ~~~~~~~ \tanh(u) = \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}, \] where the transcendental number \(e\) = 2.71828 which is discussed in The Base of the Natural Logarithm. From these definitions we can describe exponential growth or decay in terms of these functions:

\[ e^{u} = \cosh u + \sinh u; ~~~~~~~~~ e^{-u} = \cosh u - \sinh u. \] Below are graphs of the hyperbolic sine, cosine, and tangent functions:

9.17.2 Standard Trigonometric Functions

These two sets of functions – standard trig functions and hyperbolic trig functions – can be connected by using complex numbers. In particular, we can express standard trigonometric functions in an analogous fashion:

\[ \sin\theta = \frac{e^{i\theta}-e^{-i\theta}}{2i}; ~~~~~~~ \cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2}; ~~~~~~~ \tan\theta = \frac{e^{i\theta}-e^{-i\theta}}{e^{i\theta}+e^{-i\theta}}. \] As with the hyperbolic functions, we can easily see that

\[ e^{i\theta} = \cos \theta + i\sin \theta. \]

Taking this result and multiplying through by a “radius” \(r\),

\[ re^{i\theta} = r\cos \theta + i(r\sin \theta) \equiv x + iy. \] The notion of a vector appears. The above relationship describes a vector quantity which has an amplitude \(r\) at a phase angle \(\theta\). If the amplitude remains constant, but the angle changes uniformly with time, then a steady “oscillation” can be represented by this complex form.

In the above, we say that the “real” component of the complex number is \(x\), while the “imaginary” component, \(iy\), is a real number \(y\) multiplied by \(i=\sqrt{-1}\). The values \(x\) and \(y\) are the Cartesian coordinates of the vector \(z = re^{i\theta}\) in the complex plane, as shown here:

Hence, given the amplitude and phase angle \(r\) and \(\theta\) of the complex number \(z = re^{i\theta}\), we can compute its real and imaginary parts using \(x=r\cos\theta\) and \(y=r\sin\theta\). Likewise, given \(x\) and \(y\), one can find the phase angle and the amplitude: \(\theta = \tan^{-1}(y/x)\) and \(r = \sqrt{x^2 + y^2} = y/\sin\theta\). Such calculations were discussed earlier in Section Trigonometric Scales.

Some slide rules were advertised to be vector rules due to the fact that they allowed for easy conversions between Cartesian and polar coordinate systems, or allowed for more straightforward computations of the sides or hypotenuse of a right triangle. However, these operations do not involve hyperbolic functions. It was for the more-direct calculation of trigonometric functions of complex variables that brought about the invention of the true vector slide rule, as will be discussed next.

9.17.3 Functions of Complex Numbers

We are now in a position to discuss trigonometric functions of complex numbers. Trig functions of real numbers are discussed in Trigonometric Scales. The result of applying a trig function to a real number will be another real number. On the other hand, applying a trig function to a complex number in general will result in another complex number. For example, the sine of a purely imaginary number can be computed as follows:

\[ \sin(iy) = \frac{e^{i\times iy}-e^{-i\times iy}}{2i} = \frac{e^{-y}-e^{y}}{2i} = -\frac{e^{y}-e^{-y}}{2i} = i\sinh y. \]

In similar fashion one finds \(\cos(iy) = \cosh y\) (a real number), and \(\tan(iy) = i\tanh y\).

Now let’s compute the sine of a general complex number \(z_0=x_0+iy_0\) in a similar way:

\[ \sin z_0 =\sin(x_0+iy_0) = \frac{e^{ix_0-y_0}-e^{-ix_0+y_0}}{2i} = \frac{e^{ix_0}e^{-y_0}-e^{-ix_0}e^{y_0}}{2i} \\ = \frac{(\cos x_0+i\sin x_0)e^{-y_0}-(\cos x_0-i\sin x_0)e^{y_0}}{2i} \\ = \frac{-(e^{y_0}-e^{-y_0})\cos x_0 +i\sin x_0(e^{y_0}+e^{-y_0})}{2i}\\ = \sin x_0 \cosh y_0 +i \cos x_0 \sinh y_0. \] So given a complex number \(z_0 = x_0 + i y_0\), taking the sine of \(z_0\) will yield \[ z = x + iy = \sin z_0 = \sin x_0 \cosh y_0 +i \cos x_0 \sinh y_0. \]

We see that by having access to tables or scales of standard hyperbolic trig functions, one can readily compute complex values of trig functions with complex arguments. Slide rules with scales of hyperbolic trig functions are called Vector slide rules and are used for such calculations.

As another example, let’s find the hyperbolic sine of our general complex number \(z_0\):

\[ \sinh z_0 = \sinh (x_0+iy_0) = \frac{e^{x_0+iy_0}-e^{-x_0-iy_0}}{2} = \frac{e^{x_0}e^{iy_0}-e^{-x_0}e^{-iy_0}}{2} \\ = \frac{e^{x_0}(\cos y_0+i\sin y_0)-e^{-x_0}(\cos y_0-i\sin y_0)}{2} \\ = \frac{(e^{x_0}-e^{-x_0})\cos y_0+i(e^{x_0}+e^{-x_0})\sin y_0}{2} \\ = \sinh x_0\cos y_0 + i\cosh x_0\sin y_0 \] or, \[ z = x + iy = \sinh z_0 = \sinh x_0\cos y_0 + i\cosh x_0\sin y_0. \] Again, we see that the result involves products of standard trig functions and hyperbolic trig functions. Naturally, similar expressions can be found for \(\cos\), \(\cosh\), \(\tan\), and \(\tanh\) as well. Here is a table of the results:

If \(~~z\) = \(x+iy~~\),
\(\sin(z)\) = \(\sin x \cosh y + i \cos x \sinh y\)
\(\cos(z)\) = \(\cos x \cosh y − i \sin x \sinh y\)
\(\tan(z)\) = \((\sin 2x+ i \sinh 2y) / (\cos 2x + \cosh 2y)\)
\(\sinh(z)\) = \(\sinh x \cos y + i \cosh x \sin y\)
\(\cosh(z)\) = \(\cosh x \cos y + i \sinh x \sin y\)
\(\tanh(z)\) = \((\sinh 2x + i \sin 2y) / (\cosh 2x + \cos 2y)\)

To get a better grip on the type of computations we are discussing, let’s do a computer calculation. Using R we can define variables with complex values and find values of functions of these variables. For example,

# Define value of z0 (a complex number):
z0 = 1.0+1.5i
# Find the sine of this number directly:
## [1] 1.979484+1.150455i
# Now, perform the calculation in steps...
# Select the Real, Complex components of the original z0:
x0 = Re(z0)
y0 = Im(z0)
# Individually Compute and Compare the 
#    Real and Complex components of sin(z0):
## [1] 1.979484
## [1] 1.150455

Here is a plot of the calculation just performed:

If we think of a complex number \(z\) in terms of a vector, applying a function like \(\sin z\) or \(\cosh z\) will produce a new vector, with perhaps a new amplitude and phase angle. With hyperbolic trig scales on the slide rule, and appropriately placed regular trigonometric scales, the components of the results of complex vector calculations can be performed expediently on the rule.

For fun, below we make a plot of \(\sinh z\) for a small collection of complex values \(z\). Here, we just plot the initial \((x_0,y_0)\) (“o”’s) and final \((x,y)\) (“+”’s) points in the complex plane, rather than draw arrows from the origin:

z  = seq(-2,2,by=0.25)*(1+1.25i)
y  = sinh(z)

9.17.4 A Vector Calculation on a Slide Rule

Let’s take one of our points in the plot above and perform the calculation of \(\sinh z_0\) on a slide rule. We’ll use the Pickett Model 4-ES for our calculation. Suppose \(z_0 = 1 + 1.25i\). The result we are looking for is \(z = \sinh z_0 = \sinh 1\cos 1.25 + i\cosh 1\sin 1.25\). Most rules, including the one we are using here, expresses the arguments of the standard trig functions in units of degrees rather than radians, so we must convert the \(\sin\) and \(\cos\) arguments above accordingly. On the 4-ES there is a Gauge MarkR” at 5.73 on the C/D scales – 57.3 degrees per radian – so we will need to use this feature. Thus, we set up the problem as follows. To get the real component of the final vector, we will use

\[ \sinh 1\cos 1.25 = \sinh 1 \times \cos (1.25 \times 57.3^\circ ) \] To perform the calculation here are the steps that I used:

  • Line up the cursor on 1.25; line up the “1” under the cursor; slide cursor to the R gauge mark; read off degrees (= 71.6);

  • Reset the slide; read cos(71.6) = 0.315 using cursor;

  • Align left side of slide w/ cursor; move cursor to SH of 1.0;

  • Read answer on D scale: 3.71

  • Think about the numbers a moment to realize that it is actually 0.371: \(~~~ x = {\rm Re}(z) =\) 0.371.

Next, to get the imaginary component of the final vector, we use

\[ \cosh 1\sin 1.25 = \sinh 1/\tanh 1 \times \sin (1.25 \times 57.3^\circ ) \] and the steps I used are:

  • Reset the slide; align cursor to sin(71.6) = 0.95;

  • Align TH 1.0 w/ cursor

  • Move cursor to SH of 1.0;

  • Read answer on D scale: 1.46

  • Think about the numbers a moment to realize that it actually is 1.46: \(~~~y = {\rm Im}(z) =\) 1.46.

Hence, the final answer is: \(~~\sinh(1+ 1.25i)\) = \(0.371+1.46i\).


Compare with a computer-generated calculation:

## [1] 0.370567+1.46436i

Now imagine making our last plot above with 17 total evaluations using a slide rule, compared to using our modern laptop computers!

9.17.5 Polar Coordinates

The result of taking vector \(z\) and performing a function on \(z\) yields a new vector – equivalently, it takes the vector and rotates it plus affects its length. On some vector slide rules, the layout of the appropriate scales on the stock and slide of the rule allow for more direct conversions from the polar coordinates (\(r\), \(\theta\)) of the first vector to the polar coordinates of the final vector. This was an invention of Mendell Weinbach, USA, in the late 1920s, which was first implemented on the K&E Model 4093-3, and later the Models 4083-3 and 4083-5. Other manufacturers soon followed with their own models of vector rules.

Earlier we saw that \(\sinh(x_0+iy_0)\) = \(\sinh x_0 \cos y_0 + i \cosh x_0 \sin y_0\). If we wish to use polar notation \(z=\sinh z_0 = re^{i\theta}\), then we see that \[ \tan\theta = y/x = (\cosh x_0 \sin y_0)/(\sinh x_0 \cos y_0 ) = \tan y_0/\tanh x_0. \] Then, knowing \(\theta\), we can solve for \(r\): \[ r^2 = \sinh^2 x_0 \cos^2 y_0 + \cosh^2x_0\sin^2y_0 \\ = \sinh^2 x_0 \cos^2 y_0\left[1+(\cosh^2x_0/\sinh^2x_0)(\sin^2y_0/\cos^2y_0)\right] \\ = \sinh^2 x_0 \cos^2 y_0\left[1+\tan^2y_0/\tanh^2x_0\right] = \sinh^2x_0\cos^2y_0(1+\tan^2\theta) \\ = \sinh^2x_0\cos^2y_0/\cos^2\theta, \] or, \[ r = \sinh x_0\cos y_0/\cos \theta. \]

To summarize our current example, given an initial complex number \(z_0 = r_0e^{i\theta_0}\), the hyperbolic sine of \(z_0\) is computed through the following steps: \[ x_0 = r_0 \cos\theta_0 \\ y_0 = r_0 \sin\theta_0 \] then \(\sinh z_0 = re^{i\theta}\), where \[ \theta = \tan^{-1}(\tan y_0/\tanh x_0), \\ r = \sinh x_0\cos y_0/\cos \theta. \] Having hyperbolic scales and the standard trig scales on the slide of a duplex slide rule can often greatly reduce the number of steps in the calculation. Similar methods and steps can be followed to compute other trig and hyperbolic trig functions of complex numbers as were performed above. To do so in an optimal way with a particular rule, one may need the instructions that came with it.

A slide rule with appropriately placed hyperbolic and standard trigonometric scales can readily compute trig and hyperbolic trig functions of complex numbers in a relatively small number of steps. Prior to these “Vector Log-Log” rules, computing such numbers from tables and standard slide rules could take many times longer than could be done with the new slide rules with scales of hyperbolic trig functions.