Trigonometric calculations (sine and tangent)

This applies to rules with S, T and ST scales.

Introduction

This section covers the use of slide rules for the calculation of both sin and tan. The reason for doing this is that below about 5° both the sine and the tangent are effectively equal to each other. Many rules make use of this by having separate sin (S) and tan (T) scales for the higher range of angles (greater than 5.74°) and a combined sin and tan scale (ST) for lower angles. The value of 5.74° is used as sin 5.74° equals 0.1 and therefore corresponds to the end of the scale. Some rules even dispense with the lower scale as, for small angles, sin »  tan »  0.01745 x angle, to within the accuracy of a slide rule. Most, but not all rules which do this have a gauge point on the rule to simplify this calculation.

From values of sin and tan it is possible to calculate values of other trigonometric function, such as cos. As the aim of this site is to describe how to use the scales of a slide rule, not to teach mathematics, no description of how to do this is given.

The sin and tan scales are found in different places on the rule and the following sections describe how they are used in different configurations.

Sin and tan on face of stock

This is the simplest configuration. To find the sin or tan of an angle:

• Move the cursor to the angle on the S, T or ST scale..
• Read the value of the function on the D scale.

A variant of the above has the sin and tan scales on the edge of the stock and a cursor specially shaped to allow for this.

The following example show the value Tan 10°, 0.176.

Sin and tan on back of slide

This is a very common position for these scales. There are slight differences between different rules but in association with these scales there will be either a small fixed cursor or a cut out window with small mark on it on the back of the stock. To find the sin or tan of an angle:

• Move the slide to position the angle on the S, T or ST scale against the mark.
• Read the value of the function on the C scale against the 10 on the D scale.

A variant on the above has marks for sin and tan at different ends of the rule. In which case use the 10 or the 1 on the C scale as appropriate.

The following is from a rule with with mark for Tan, T scale,  at one end and two marks for Sin Tan (ST scale) and sin (S scale) at the other. The rule shows both faces for tan 15°, 0.267

Other variants

A further variant has sin values from 35' to 90° and the A and B scales are used rather than the C and D. Some rules have the angles in decimals of a degree and others in minutes. More details with sample calculations are given here.

Thornton rules (and those of PIC an associated company) have a patented method which enables the values to be given in less space but require more movements of the slider.