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A slide rule can find sine, cosine and tangent without a calculator. Put the cursor on an angle printed on the rule’s trigonometric scale, read the function on the adjacent C or D scale, and then use the ordinary logarithmic scales for multiplication or division. The result is an engineering approximation, so you choose the decimal placement and check any result that requires more precision than the rule can show.
What the S, T and ST scales do
Scientific slide rules add trigonometric graduations to the familiar C and D logarithmic scales. The lettering is usually:
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| Scale | Function | Typical use |
|---|---|---|
| S | Sine (and often cosine on a reverse or contrasting-color graduation) | Angles from roughly 5.8° through 90° |
| T, T1 or T2 | Tangent | Medium and large angles; the exact ranges depend on the rule |
| ST | Small-angle sine and tangent | Approximately 0.58° to 5.8° on the layout described by the Oughtred Society manual |
Those ranges are approximate, not a universal standard. Duplex rules may split trigonometric scales across opposite faces, while single-sided rules may place them above or below C and D. Before calculating, identify the scale labels and angle units on your particular model.
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Basic procedure
- Locate the angle in degrees on the S scale. Degree marks may run in the opposite direction from the ordinary C and D scales.
- Move the cursor hairline to that angle.
- Read the corresponding number on the C scale (or the paired scale specified by your rule’s instructions). This is the sine value.
- Place the decimal point from the expected range. A sine is between 0 and 1 for angles from 0° to 90°.
For example, the Oughtred Society manual’s worked values are sin 30° = 0.5 and sin 60° = 0.866. The extra digits are only as reliable as the scale length and your ability to interpolate between graduations.
Small angles
For angles below the main S-scale range, use ST if your rule provides it. The ST graduations are expanded for small-angle work, where the ordinary S scale would crowd the marks together. Because layouts differ, follow the printed angle direction and the model’s instruction sheet rather than assuming every ST scale starts at the same mark.
Finding a cosine with the sine scale
Cosine is obtained from the complementary angle:
cos θ = sin (90° − θ)
Thus sin 30° equals cos 60°. On many slide rules, cosine values are printed in reverse order or in a contrasting color on the S scale. To find cos 30°, either read the cosine graduation at 30° if your rule has one, or move the cursor to 60° on the sine graduation and read the value. The result should be about 0.866.
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Check that you are using degrees, not radians. Most traditional trig scales are marked in degrees; a radian input requires conversion before you set the cursor.
Finding a tangent
Using T, T1 or T2
- Identify the tangent scale your rule labels T, T1, T2 or an equivalent designation.
- Set the cursor to the angle on that scale, observing whether the graduations increase from left to right or right to left.
- Read the tangent value against the paired C or D scale specified for the rule.
- Supply the decimal magnitude. Tangent grows rapidly as the angle approaches 90°, so confirm that the scale’s range matches your angle.
Some instruments divide tangent into T1 and T2 sections to keep the graduations readable over different angle ranges. Others combine ranges or print tangent on a duplex face. There is no single placement or range shared by every model, which is why the scale legend matters more than the letter alone.
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When to use ST for tangent
For a small angle, use the ST section when available. It is designed for the narrow-angle region in which sine and tangent are close and the main trigonometric scales have limited resolution. If your rule has no ST scale, you can still obtain an estimate from its T scale, but the reading may be less distinct.
Combining a trig value with multiplication or division
The trigonometric scale supplies a factor; C and D perform the arithmetic. To evaluate 15 × sin 30°:
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- Set the cursor at 30° on S.
- Read sin 30° as 0.5 on the paired C scale.
- Use the normal slide-rule multiplication setup: align 1 on C with 15 on D (or use the appropriate index and scale direction for your rule).
- Read the product at the cursor position on D. The result is 7.5.
For a larger expression such as 24 cos 35° ÷ 3, obtain cos 35° from the complementary sine graduation, then perform the multiplication and division with C, CI and D as you would for any non-trigonometric factors. Keep track of powers of ten separately; the rule displays significant digits, not the decimal exponent.
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- Significant figures: A standard slide rule commonly supports only a few meaningful digits. Read the finest graduation you can and interpolate cautiously.
- Magnitude: Decide the decimal position from bounds or a rough mental estimate. For example, sin 30° must be 0.5, not 5.
- Angle reading: A small cursor-placement error changes the result, especially on tangent near 90°.
- Verification: Recheck safety-critical, financial, scientific or precision-sensitive calculations with a higher-precision method.
These limits are inherent to an analog logarithmic instrument, not a malfunction. A longer rule generally offers finer graduations, while a duplex layout can provide more scale length and more trig ranges at the cost of a steeper learning curve.
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Choosing a slide rule for trigonometry
If trig work is your priority, compare the instrument on practical features rather than brand name:
| Comparison point | Why it matters |
|---|---|
| S, T and ST availability | Provides direct sine, tangent and small-angle readings instead of relying on approximations. |
| Angle notation and range | Confirms whether your angles fit the printed graduations and whether degrees are clearly marked. |
| Cursor readability | A fine, stable hairline improves interpolation and reduces parallax. |
| Single-sided versus duplex layout | Single-sided rules are easier to learn; duplex rules can offer longer or additional trig scales. |
| Scale length | Longer scales spread graduations farther apart, improving practical significant-figure resolution. |
A scientific slide rule with clearly labeled S, T and ST scales is the most flexible choice. A printed instruction manual or historical reprint can be useful when the rule’s scale arrangement is unfamiliar, because the exact C/D pairing and cursor procedure vary by model.
Why trig scales mattered historically
Smithsonian describes slide rules as analog computing devices built from linear and logarithmic scales. By around 1900, some models included logarithmic trigonometric scales and had become standard tools for scientists and engineers. They remained in everyday professional use until inexpensive electronic calculators arrived in the 1970s. Learning S, T and ST therefore shows both the mathematics of trigonometric functions and why a compact mechanical instrument could handle engineering calculations before electronic displays.
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