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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware matchOn a scientific slide rule, the S scale gives sine, the complementary graduations give cosine, and T scales give tangent. The ST section handles very small angles. Put the cursor on the angle, read the trigonometric value against the C or D scale, and then use the ordinary C/D scales for multiplication or division.
What the S, T and ST scales do
Trig scales turn a slide rule from a logarithmic multiplication device into a compact analog calculator for common trigonometric functions. Their markings are not identical on every model, so identify the letters, colors and angle convention on your particular rule before calculating.
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| Scale | Primary function | Typical coverage or use |
|---|---|---|
| S | Sine | Approximately 5.8° to 90° on the layout described by the Oughtred Society manual |
| Reverse or contrasting S graduations | Cosine through complementary angles | Use sin X = cos (90° − X) |
| T, T1 or T2 | Tangent | Range and graduation design vary by model |
| ST | Small-angle sine and tangent work | Approximately 0.58° to 5.8° on the Oughtred Society layout |
The stated angle spans are approximate, not a universal standard. Duplex rules may place these scales on different faces, while single-sided rules may combine or color-code them differently.
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Calculating sine
Basic procedure
- Find the angle on the S scale. Confirm whether the scale increases from left to right or is reversed on your model.
- Place the cursor hairline exactly over the angle marking.
- Read the corresponding number on the C scale (or the D scale when your rule’s layout uses D for the direct reading).
- Supply the decimal magnitude yourself. Slide-rule scales indicate significant figures, not the position of every zero.
Using the Oughtred Society manual’s examples, placing the cursor at 30° gives sin 30° = 0.5; placing it at 60° gives approximately sin 60° = 0.866. The last digits are an estimate from the graduations, not a calculator-style exact display.
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Angles below the main S range
For an angle in the small-angle region, move to the ST section when your rule provides one. On the cited layout, ST covers roughly 0.58° through 5.8°. The exact transition and labeling differ among manufacturers, so do not assume that a mark labelled “5” means the same unit or position on every instrument.
Finding cosine with the complementary angle
Cosine is obtained from the identity cos θ = sin (90° − θ). For example, cos 60° equals sin 30°, so the S-scale reading for 30° gives 0.5. Many rules print cosine graduations in reverse or in a contrasting color on the same trig scale. If those markings are present, you can read cosine directly; otherwise, subtract the angle from 90° and use S.
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- Convert the requested angle to its complement: 90° − θ.
- Locate that complementary angle on S (or on the cosine graduations if your rule provides them).
- Read the value against the appropriate C or D scale.
This method avoids treating the reverse numbering as a second independent mathematical scale; it is simply the complementary-angle relationship printed for faster work.
Calculating tangent
Using T scales
Locate the scale labelled T, T1, T2 or an equivalent tangent designation, then place the cursor over the angle and read against the paired C/D scale. Rules divide tangent values into sections because tangent changes rapidly near 90°. The section names, ranges and whether the graduations are on the front or back differ by model.
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- Read the rule’s instruction or scale legend to determine which T section contains your angle.
- Set the cursor on the angle mark in that section.
- Read the tangent value on the C or D family indicated by the rule’s layout.
- For a very small angle, check ST first; it may provide a more legible and appropriately spaced reading.
Do not transfer a T-scale range from one rule to another without checking the legends. A duplex engineering rule can place T1 and T2 on opposite faces, while another instrument may use a single T scale with a different angular span.
Combining a trig value with multiplication or division
The cursor supplies the trigonometric value; the slide and its logarithmic C/D scales perform the arithmetic around it. To evaluate 15 × sin 30°, first put the cursor on 30° on S. The trig reading is 0.5. Then use the normal multiplication setup: align 1 on C with 15 on D, move the cursor to 5 on C, and read 7.5 on D. On some rules the trig operation and multiplication are integrated by reading the trig value on D after setting the appropriate C or CI index; follow the instrument’s printed instructions.
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The same principle handles expressions such as A ÷ tan θ: obtain tan θ from T, then use the C/D scales for the division. Keep the decimal placement separate from the scale reading. If the trig scale gives 0.866 and the arithmetic scale gives 3.2, the result is about 2.77, with the meaningful precision limited by the graduations and the inputs.
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- Slide rules provide approximate significant figures; they do not display an exact decimal expansion.
- Estimate between graduations only as far as the scale length and cursor line remain readable.
- Determine the decimal magnitude from the mathematical context. A slide rule can show the digits of 0.5 but cannot decide whether a result should be 0.5, 5 or 50 without your setup.
- For safety-critical, financial, surveying or otherwise precision-sensitive work, verify the result with an independent calculation.
Longer scales generally make interpolation easier, but readability also depends on cursor quality, hairline thickness and printing. A duplex rule may offer more functions while requiring you to turn the instrument over during a calculation.
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How to choose a rule for trig work
| Comparison point | Why it matters |
|---|---|
| S, T and ST availability | Confirms that sine, tangent and small-angle work are supported without external tables. |
| Angle notation and range | Reveals whether the rule uses degrees and which T or ST sections cover your angles. |
| Cursor readability | A sharp, stable hairline improves interpolation and reduces parallax errors. |
| Single-sided versus duplex layout | Single-sided rules are simpler to learn; duplex rules can provide more scales but may require flipping. |
| Scale length | Longer scales usually permit finer visual estimation of significant figures. |
A scientific slide rule with clearly labelled S, T and ST scales is the practical choice for regular trigonometric work. An instruction manual or historical reprint can be useful when the rule’s layout is unfamiliar, particularly for duplex models whose index and cursor conventions differ.
Why these scales mattered before calculators
The Smithsonian describes slide rules as analog computing devices built from linear and logarithmic scales. By around 1900, some models added logarithmic trigonometric scales and became standard tools for scientists and engineers. They remained in widespread professional use until inexpensive electronic calculators arrived in the 1970s.
The trig scales explain that historical role: one instrument combined multiplication, division, powers and common angle functions without batteries or a table of printed values. The trade-off was visible precision. Engineers accepted a few reliable significant figures because the calculations were fast, portable and repeatable.
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