Slide Rule Mania Part 2

Today I got the idea to converse about Tangents, and the calculations associated with determining the values, including angles for right and obtuse triangles.

Starting off with an easier road to calculating the values, we will delve into Right Triangles, where we will use sines, cosines, tangents and cotangents. Again we are using the Post Versalog 1460 slide rule for calculations. It has scales for the above labelled S, Cos, ST, SecT and T. All these scales are on the slide, and are used with (usually) the C and D scales.

So, let’s jump right in, shall we?

Right triangle.

We already know the C value, but need to determine the remaining angles, and the lengths of the three sides. We will read the S or T scales.¹ The a is also known as opposite, the b as adjacent, and c as hypotenuse. We can use many combinations of known values, as such,

\[\sin{\theta} = \frac{a}{c}, \ \tan{\theta} = \frac{a}{b}, \ \cot{\theta} = \frac{b}{a}, \ \cos{\theta} = \frac{b}{c}\]

If we know a and b, c is Pythagoras’ Theorem, \(c = \sqrt{a^{2} + b^{2}}\). Otherwise we can use,

\[b = c \cos{A} = a \cot{A} = c \cot{A} = c \sin{B} = a \tan{B}\]

and,

\[a = c \sin{A} = b \tan{A} = c \cos{B} = b \cot{B}\]

Let’s say we have A (\(\theta\)) = 35^o, b = 15, and we need to determine a, c and angle B. We can use \(a = \tan{35} \times b\). Set index to 15, hairline (HL) to 35 on the T scale, and read a = 10.5. Now we have a and b, and can calculate c. Normally we could use the A and B scales, but the Post Versalog 1460 has none. So we can use the R scales, as in this post for impedance. Set C scale index (try left first) over smaller number (10.5) on R1 scale, slide HL to 15 on R1 scale, and read 2.04 on C scale. Add 1 for 3.04, and read R1 scale for c = 18.3. That may seem complicated, but it takes longer to talk about it than do it!

The remaining angle is simply found by 180^o - 90^o - 35^o = 55^o, as all right triangles’ inner angles total 180^o.

Perhaps we have a surveying situation where we must determine a distance from point A to point C, but there is a river or canyon between which we cannot navigate. We can take other measurements and calculate the missing distance. The below diagram is an obtuse triangle.

Survey measurement.

So, we have turned an angle from point A to B as 73.3^o, then moved to point B and turned the angle from point A to point C as 101^o. Then C angle is 180^o - 73.3^o - 101^o = 5.7^o. We measured the distance from A to B as c = 54 feet. Note the above diagram is not to scale.

We can determine b using the law of sines. To use the law of sines, we must have a known side opposite a known angle.

\[\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}\]

We know A, B and C angles, and c distance, so we use \(\frac{\sin{C}}{c}\) and \(\frac{\sin{B}}{b}\). C is a small angle, so we use the ST scale. D and C scale readings will be between 0.01 to 0.1.² So, align the scales and set 5.7 on the ST scale, then move the slide to set 54 under the HL, and read 0.00184 on the D scale at the index. As B > 90^o, we subtract 180^o - 101^o = 79^o. Align the scales, move the HL to sine of 79^o, set 0.00184 (C scale) under the HL, read b = 533.7 feet on the D scale at the index.

If we wished to continue and find the final unknown, the a distance, we use the same procedure, \(\frac{\sin{A}}{a}\). Align scales, move the HL to sin 73.3^o, move the slide to set 0.00184 under the HL, read 520.8 feet on the D scale at the index.

Just a side note, the above diagrams were drawn on the reMarkable 2 tablet, and displayed on a computer using a whiteboard program. Then a screenshot was taken and edited using GIMP, and finally saved as an image. See this post for more info.

And that’s all for these ramblings. Have a great day in the Lord Jesus Christ, The Word.³ Once again, Happy Holidays, and God Bless you and yours!

Footnotes

1. If the tan > 45 and cot < 45, we would read the CI scale.

2. Each decade adds another zero to results when using the ST scale.

3. First John 1:1 That which was from the beginning, which we have heard, which we have seen with our eyes, which we have looked upon, and our hands have handled, of the Word of life.