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Many E-6B flight computers include a trig scale. On mine it's marked 
DRIFT CORR. This scale is equivalent to the tangent scale on a slide 
rule. That is, when the pointer is set to degrees on the drift scale, 
the tangent of that angle is read on the outer main scale of the E-6B, 
opposite the index (1.0 mark) on the disc.

Just for fun, I will attempt a celestial sight reduction (intercept
method) with only that scale. And for even more fun, I will use a dead 
reckoning position which does not give nice integer angles. All angles 
will require interpolation.

The problem: lat = N40 41, local hour angle = 37 29, declination = -23
25. Calculate altitude and azimuth.

First, minutes can be converted to decimal degrees at a single disc
setting. Put 6 on the inner main scale opposite 1.0 on the outer. Then
opposite minutes on the inner scale find decimal degrees on the outer.
Lat = N40.68, LHA = 37.48, dec = -23.42.

I will use a quasi Bygrave solution. A few years ago I ran a computer
Monte Carlo simulation of a 10 inch slide rule working the Bygrave 
formulas. Accuracy was better than I expected.

http://fer3.com/arc/m2.aspx/Bygrave-formula-accuracy-on-10-inch-slide-rule-Hirose-jul-2009-g8985

However, an E-6B is made for robustness, not high accuracy. And the
position of its tangent scales - near the center of the disc - could
hardly be worse. I'm not optimistic, but let's see what happens.

First calculate y = arctan(tan dec / cos LHA) = arctan(tan -23.4 / cos 37.5)

tan 23.4 = .434 (ignore the negative sign on dec.)

Since the E-6B has only a tangent scale, we have a problem with cos 
37.5. However, there is a trigonometric identity to convert a tangent to 
a secant or vice versa. In this case, it gives 1.26 for sec LHA. Note 
that the formula for y divides by cos LHA. That's equivalent to 
multiplication by sec LHA. So we simply muliply by tan dec (.434) and 
sec LHA (1.26) and get .547. Set the inner index to .547, read 28.8° on 
the tangent scale. That's y.

The value Y (note capital letter) is 90° - lat - y = 20.5°.

Azimuth = arc tan (tan LHA * cos y / cos Y) = arc tan (tan 37.5 * cos
28.8 / cos 20.5). The tangent is easy. For cos 28.8, from the y formula
we know tan 28.8 = .547. Use the identity formula to obtain the secant:
1.14. Take the reciprocal to get .877 = cos y.

Similarly, cos 20.5 is found via its tangent and the identity formula. I
get 1.068 for sec and .936 for cos 20.5.

Therefore, azimuth = arc tan (tan 37.5 * .877 / .936) = arc tan .720 = 
35.7°. Add 180° because LHA <= 180°. Final azimuth = 215.7°

Altitude = arc tan (cos az * tan Y). For cos az, recall that tan az was 
.720. The identity formula converts that to cos az: .811. Therefore, 
altitude = arc tan(tan Y * .811) = 16.8°. Now let's see how close I got.

Reduction by calculator gives az = 215.71° (vs. 215.7), altitude = 
17.05° (vs. 16.8). That's pretty good, given the coarse scales and 
complicated indirect methods I had to use.


There are actually two concentric tangent scales on an E-6B, that is, 
the ones that have the scales. Some don't. This E-6B has them. They are 
the innermost scales on the disc, and share a common window.


http://www.aprindustries.com/index.php/E6-B-Flight-Computers/E6-B2WHL-Top-of-the-Line-Universal-high/low-cursor-arm-computer/flypage.tpl.html

(I don't know if the features on this computer are worthy of the 
company's bragging. But it's interesting that inventors are still 
patenting improvements on the E-6B even in this age.)

The inner scale goes from 7.5° to 45°, the outer from 1° to 7.5°. Two 
scales are necessary because the tangent function covers almost two 
decades from 1° (tan = .017) to 45° (tan = 1.00). The small angle scale 
isn't essential, since the tangent of a small angle is practically equal 
to the angle itself in radians.

Test that approximation on tan 3°. Convert 3° to radians: opposite 3 on 
the outer scale, set 57.3 on the inner. Read .0525 opposite the inner 
index. Calculator says .05241.

A variation of that method gives tangents of angles beyond the range of 
the scale. For example, what is tan 87°? The same as 1 / tan 3°. 
Assuming the disc is still set to tan 3°, read 19.06 opposite the 
*outer* index.

The principle is that whatever value the inner index points to, the 
outer index points to its reciprocal. For example, if you set the inner 
index to .5, the outer index is at 2. (This is equally true of the C and 
D indices on an ordinary slide rule.) If you set an angle on the E-6B 
tangent scale, its tangent is at the inner index, and simultaneously its 
cotangent is at the outer index.

Furthermore, the squares of tan and cot are obtainable without moving 
the disc. For example, set 25° on the tangent scale. At the outer index 
read cot 25° = 2.152. On the outer scale find 2.152 and read cot² 25° = 
4.64 on the inner scale. (After my calculator said 4.60, I tried again. 
The result was even worse. Apparently my E-6B is just inaccurate here.)

The squares of tan and cot are important because of these identities:

sec² x = 1 + tan² x
csc² x = 1 + cot² x

Since secant = 1/cosine and cosecant = 1/sine, we can get sines and 
cosines from the tangent scale - if square roots can be extracted.

On the E-6B, that's done on the main scales by trial and error. For 
example, take the square root of 5.5. First put your thumb on the outer 
scale at 5.5. Set the inner index about halfway (clockwise) from the 
outer index to your thumb. It looks like 2.5 is a reasonable first 
guess. Find 2.5 on the inner scale, see if it's opposite 5.5 on the 
outer. Oops, too much. Rotate the disc to remove about half the error. 
That puts the inner index at about 2.32. Make it exactly so, find 2.32 
on the inner scale, see if it's opposite 5.5. Little too small this 
time. After one more correction, 2.34 looks almost perfect.

This is easier if all your estimates - except the final refinement - 
fall exactly on a graduation, so no interpolation is needed. And if your 
eyes are not young, strong reading glasses and a pencil point to serve 
as a substitute cursor will help.

I don't recommend any of this unless you're looking for a challenge! 
Throughout the calculation I checked myself with a conventional 10 inch 
slide rule, and found it much faster and more accurate. It also 
prevented a couple blunders. Nevertheless, in an emergency the aviation 
computer can give you sine, cosine, tangent, cosecant, secant, 
cotangent, and the inverses of those functions, for any angle 0 - 360°, 
accurately enough to cross an ocean.

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