How to Use S, T, and ST Trig Scales on a Slide Rule

This guide explains how to use the S, T, and ST scales found on many slide rules to compute sines, cosines, tangents, and their inverses for angles in degrees. You will learn to read these scales, set the cursor, and interpret the results, whether you are solving trigonometry problems, converting angles to ratios, or working with small angles. No prior slide rule experience is needed beyond basic multiplication.

What You’ll Need

• A slide rule with S, T, and ST scales (most scientific and engineering rules have them).
• A magnifying glass (optional, but helpful for reading fine graduations).
• A pencil and paper to note intermediate values.

Step-by-Step

1. Identify the scales. On most rules, the S scale is on the slide or the body, marked with angles from about 5°45′ to 90°. The T scale covers roughly 5°43′ to 45°. The ST scale (sometimes labeled “SR” or “S&T”) covers small angles, typically from about 0.55° to 5.7°. The exact range varies by model, so check the markings.
2. Understand the relationship. The S scale gives sine values; the T scale gives tangent values. The ST scale is used for both sine and tangent of small angles, where they are nearly equal. The C (or D) scale is the “number” scale that pairs with them.
3. Find the angle on the S scale. For a sine, locate the angle in degrees on the S scale. For example, to find sin 30°, find the “30” on the S scale.
4. Read the sine value. Place the hairline of the cursor over that angle. Then read the value on the C scale (or D scale, depending on the rule) directly under the hairline. That number is the sine. For sin 30°, you should read approximately 0.5 on the C scale.
5. For cosine, use the complementary angle. Since cos θ = sin (90° − θ), you can find the cosine by locating (90° − θ) on the S scale. For example, cos 30° = sin 60°, so find 60° on the S scale and read the value on C—again 0.5.
6. For tangent on the T scale. Locate the angle on the T scale. For tan 30°, find “30” on T. Read the value on C: you should get about 0.577. Note that the T scale is typically used for angles from about 5.7° to 45°; for angles above 45°, use the reciprocal relationship: tan θ = 1/tan (90° − θ).
7. For small angles on ST. For angles between about 0.55° and 5.7°, use the ST scale. Locate the angle on ST and read the sine or tangent directly on C. For example, sin 3° ≈ 0.0523, and tan 3° ≈ 0.0524—both essentially the same on the slide rule.
8. Inverse functions. To find an angle from a sine value, reverse the process: set the hairline over the sine value on the C scale, then read the angle on the S scale. For example, to find arcsin 0.5, set the hairline at 0.5 on C, and read 30° on S. Similarly, for arctan, use the T scale.
9. Handle angles outside the scales. For angles less than about 0.55°, the sine and tangent are approximately equal to the angle in radians. Convert degrees to radians (multiply by π/180) and use that value. For angles greater than 90°, use the periodicity: sin θ = sin (180° − θ) for θ between 90° and 180°, and tan θ repeats every 180°.
10. Practice with a known value. Try sin 45°: set hairline on 45° on S, read on C—you should get 0.707. Then try tan 45°: you should get 1.0. These checks confirm you are reading the correct scale.

Common Mistakes

• Using the wrong scale. The S scale is for sine, not tangent. Mixing them up yields incorrect results. Always verify which scale you are on.
• Reading the C scale instead of D. On some rules, the C scale is on the slide and D is on the body. When the slide is centered, they match, but if the slide is moved, they differ. For trig functions, the slide is usually set so that C and D align, but always check the alignment.
• Forgetting the complementary angle for cosine. The S scale does not directly show cosines; you must subtract from 90°. A common error is to read the angle directly as a cosine.
• Ignoring the ST scale for small angles. Using the T scale for a 2° angle will give a wrong value because 2° is outside T’s range. Use ST.
• Misreading the angle on the ST scale. The ST scale is compressed; small misalignments cause large errors. Use the magnifying glass and align the hairline carefully.

Tips

• Always set the cursor hairline precisely over the center of the graduation mark. A slight offset changes the result.
• When reading the value on C, note the decimal point. The C scale typically runs from 1 to 10, so a reading of 5 means 0.5, 5.0, or 50 depending on the context. For sine, the result is between 0 and 1, so 5 on C means 0.5.
• For angles in degrees, ensure your rule is set to degrees, not radians. Some rules have separate S and T scales for radians, but most use degrees.
• Memorize common values (sin 30° = 0.5, tan 45° = 1) to check your technique.
• If your rule has a “ST” scale but also a “S” and “T” scale, use ST only for angles below about 5.7°; for larger angles, use S and T.

Troubleshooting

• Result seems off by a factor of 10. Check the decimal point. For sine, the value should be between 0 and 1. If you read 5, it is 0.5; if you read 50, you are likely on the wrong scale or misread the C scale.
• Cannot find the angle on the T scale. The T scale covers only up to 45°. For angles above 45°, use the complementary angle: tan θ = 1/tan (90° − θ). For example, tan 60° = 1/tan 30° = 1/0.577 = 1.73.
• Angle is less than 0.55°. The ST scale does not go that low. Use the approximation sin θ ≈ tan θ ≈ θ (in radians). Convert degrees to radians: θ_rad = θ_deg × π/180. For 0.5°, that gives 0.00873.
• Slide rule gives a different result than a calculator. Slide rules are accurate to about 3 significant figures. Differences in the last digit are normal. If the difference is large, recheck your scale selection and alignment.
• The ST scale seems to have two sets of numbers. Some rules have the ST scale marked with both sine and tangent values, or with angles in degrees and radians. Use the degree markings for typical problems.
• Cursor is not aligned with the scale. Ensure the cursor is clean and moves smoothly. If it is loose, tighten the screw (if adjustable) or press gently to keep it perpendicular.

Frequently Asked Questions

Q: Can I use the S scale for cosine directly?
A: No, the S scale is for sine. For cosine, use the complementary angle: cos θ = sin (90° − θ).

Q: What is the difference between T and ST scales?
A: The T scale is for tangents of angles from about 5.7° to 45°. The ST scale is for sines and tangents of small angles (typically 0.55° to 5.7°), where the two are nearly equal.

Q: How do I find the tangent of an angle greater than 45°?
A: Use the reciprocal identity: tan θ = 1/tan (90° − θ). For example, tan 60° = 1/tan 30° = 1/0.577 ≈ 1.73.

Q: What if my slide rule does not have an ST scale?
A: For small angles, use the approximation sin θ ≈ tan θ ≈ θ (in radians). For angles above about 5.7°, use the S and T scales. Some rules have a combined “S” scale that includes small angles; check your rule’s manual.

Q: Why does my slide rule give 0.5 for sin 30° but my calculator says 0.5 exactly?
A: Slide rules are analog devices with limited precision (usually 3 significant figures). 0.5 is exactly representable, but for other angles like sin 20°, you might get 0.342 instead of 0.3420—that is normal.

Q: Can I use these scales for radian measure?
A: Most slide rule trig scales are in degrees. If you need radians, convert first: multiply the radian value by 180/π to get degrees, then use the scales. Some rules have separate radian scales, but they are less common.

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