Tutorial: Trigonometric Scales

Trigonometric Scales

For tutorials on hyperbolic scales read this.
For discussion on angles in Degrees,Minutes, and Seconds, read this.
For discussion on the P,P2 scales, read this.

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Instructions
Examples

Introduction

The slide rule is a fantastic device for gaining a deeper understanding of trigonometry, since there are far too many trig functions to include as separate scales, and of the functions represented, only subsets of their complete domains are found on the scales. Over the centuries of slide rule design, all information has been condensed into the absolute minimum required to find the correct decimal approximations of these function values. Without a knowlege of what the trig functions mean, and the identities used to manipulate them, achieving the correct answer will not always be easy.

The trig scales contain degree angles from about 0.5º to 90º(85.5º for tans). The angle is found on the trig scale and the value is read off of the C/CI or D/DI scales,(as usual depending on stock or slider configuration), and adjusted by a factor of ten. The scales come in two varieties - S and T, S for Sines/Cosines, and T for Tangents/Cotangents. Both types of scales have large angle and small angle varieties to cover a wider range of angles.

Angles in radians

There are no scales with radian angles. Conversion between degrees and radians is accomplished by multiplying or dividing by the degrees to radians gauge point, (pi/180), appearing as the greek letter lowercase delta around 1.7 on C. It is 100x too large, so you'll have to adjust your answer by a factor of 100.

Angles between 0º and 0.5º

The scales don't have small angles down to zero, due to the nature of the C scales. Therefore, for angles between 0º and 0.5º, small-angle approximations are used. When we're talking about 0.5º or less, the sin or tan of the angle in radians is just the angle itself ( to more than 7 places!), and the cos of the radian angle is almost one, to five decimal or more. These approximations don't hold for degrees, only radians, so a conversion step will be necessary if degrees are required.

So why not construct trig scales where the values were read off the L scale, which runs between 0 and 1, and thus cover the entire first quadrant with one scale? The purpose of the actual arrangement is to facilitate chained calculations, not just for looking up individual sines or cosines, and one gets a lot more speed and precision if a multiplication or division can be performed on the value immediately, instead of remembering the number and transferring it to the C scale for further operations.

Angles outside the first quadrant

For angles greater than 90º or less than zero, appropriate shift and symmetry identities are employed, which may involve adding or subtracting multiples of 90, which you'll have to do in your head. There's a lot of trig identities, and more than one way of writing each one. It would be possible to boil down the relevant identities into a single formula for moving any function/angle combination into a first-quadrant form that the slide rule can handle, but it would be too esoteric for the scope of this tutorial, performing algebraic operations on function names as well as angles. Here, instead, is an incomplete list of identities you may find useful for angles not found on the slide rule:

sin(-x)= -sin(x)

cos(-x) = cos(x)

tan(-x) = -tan(x)

sin(x) = cos(90-x)

cos(x) = sin(90-x)

tan(x) = cot(90-x)

sin(x) = -sin(x-180)

cos(x) = -cos(x-180)

tan(x) = tan(x-180)

sin(x) = -cos(x-270)

cos(x) = sin(x-270)

tan(x) = -cot(x-270)

What about cos,cot,csc,sec?

The scale names might seem to imply only Sine and Tangent functions are available, but that is not the case. Both sets of scales are double scales, with two sets of angles printed on each so that Cosines and Cotangents are read side by side with Sines and Tangents. These, and the use of the CI/DI scales, also provide Cosecants and Secants, completing the list.

Inverse Functions

As with all slide rule operations, both the function and the inverse function are always available, simply by reading the other way. Thus arcsin values can be set on C and their corresponding angles (adjusted by the appropriate factor of 10) read off of S.

Instructions

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

We also use S2,T2 instead of and ST scale, but if your slide rule has an ST scale, you can use it for either S2 or T2, with slight inaccuracies.

All trig scales have a double set of numbers. By "forward" scales we mean the numbers increasing to the right, and "reverse" scales mean the numbers increasing to the left.

These instructions apply identically to decimal or DMS scales.

In all cases, we assume the angle x is between 0º and 90º. If it is outside this range, you'll have to use identities to create an equivalent expression, such as sin(110º) = cos(90º-110º)=cos(-20º)=cos(20º);

y = sin x

find x on S or S2's forward scales, set cursor
read y on C at cursor.
divide y by 10 if on S, by 100 if on S2

y = csc x

find x on S or S2's forward scales, set cursor
read y on CI at cursor.
multiply y by 10 if on S2

y = arcsin x

set cursor at coefficient of x on C
if x between 0.1 and 1, use S
if x between 0.01 and 0.1, use S2
answer at cursor on S or S2's forward scales.

y = arccsc x

set cursor at coefficient of x on CI
if x between 1 and 10, use S
if x between 10 and 100, use S2
answer at cursor on S or S2's forward scales.

y = cos x

find x on S or S2's reverse scales, set cursor
read y on C at cursor.
divide y by 10 if on S, by 100 if on S2

y = sec x

find x on S or S2's reverse scales, set cursor
read y on CI at cursor.
multiply y by 10 if on S2

y = arccos x

set cursor at coefficient of x on C
if x between 0.1 and 1, use S
if x between 0.01 and 0.1, use S2
answer at cursor on S or S2's reverse scales.

y = arcsec x

set cursor at coefficient of x on CI
if x between 1 and 10, use S
if x between 10 and 100, use S2
answer at cursor on S or S2's reverse scales.

y = tan x

x is somewhere on T or T2's forward or reverse scales. Find it and set the cursor.
if x <= 45º, read y on C

divide y by 10 if on T, by 100 if on T2

if x > 45º read y on CI

multiply y by 10 if on T2

y = cot x

xº is somewhere on T or T2's forward or reverse scales. Find it and set the cursor.
if x <= 45º read y on CI

multiply y by 10 if on T2

if x > 45º, read y on C

divide y by 10 if on T, by 100 if on T2

y = arctan x

if x < 1, set cursor at coefficient of x on C

if x < 0.1, y is at cursor on T2's forward scales
otherwise y is at cursor on T's forward scales

if x > 1 set cursor at coefficient of x on CI

if x < 10, y is at cursor on T's reverse scales
otherwise y is at cursor on T2's reverse scales

y = arccot x

if x < 1, set cursor at coefficient of x on C

if x < 0.1, y is at cursor on T2's reverse scales
otherwise y is at cursor on T's reverse scales

if x > 1 set cursor at coefficient of x on CI

if x < 10, y is at cursor on T's forward scales
otherwise y is at cursor on T2's forward scales

That's too many rules to memorize. Fortunately they're really all doing the same thing, except that csc,sec, and cot are the reciprocals of sin,cos, and tan, so the CI scale is used. The "arc" functions are the inverse functions, and the steps go backwards. If you get familiar with finding sin and tan values on the slide rule, you can derive all the other instructions from the function definitions and relationships learned in trigonometry.

Examples

sin 20º: 20º is on the S scale. (using forward scale for sines) Set cursor.; read 3.42 on C; we used S, so divide by 10: result 0.342
sin -4.2º: -4.2º isn't between 0 and 90. Convert with identities:; sin(-x) = -sin(x). So we can work with 4.2º and negate the result.; 4.2º is on the S2 scale. (using forward scale) Set cursor.; read 7.324 on C; we used S2, so divide by 100: result 0.07324; negate result as described above: -0.07324
sin 0.0065º: 0.0065º is too small for the S2 scale. Use small angle approximation for sines:; ie: Convert to radians and call it an answer:; multiply 6.5 by the delta ( degrees to radians ) gauge point, get 1.1345; adjust decimal point: *0.01 for the gauge point, * 0.001 for the angle, *10 because we changed magnitude,; answer 1.1345E-4.
cos 212º: use identity to create an equivalent expression between 0º and 90º:; cos 212º = -cos( 212º-180º) = -cos 32º; so we can work with cos 32º and negate the result.; 32º is on the S scale. (using reverse scale for cosines) Set cursor.; read 8.480 on C, divide by 10 because using S scale: 0.8480.; final answer is -0.8480 after negating the result.
tan 30º: 30º is on T, and it's less than 45º,; so we read answer on C and divide by 10.; answer 0.5774
tan 60º: 60º is on T, and it's greater than 45º,; so we read answer on CI, no adjustment necessary.; answer 1.732
arcsin 0.53: 0.53 is between 0.1 and 1, so we use S.; set cursor at 0.53 C, look up answer 32.005º on S's forward scales.
arctan 76.0: 76 > 1, so we set cursor at 7.6 on CI; 76 > 10, so we look up answer on T2's reverse scales; answer is 89.246º