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Small angles

sin 4° is about 0.0698 because 4° is 0.0698 radians. Use sin θ ≈ tan θ ≈ θ, know how far off it is at 20°, and size things from the angle they fill.

  • Geometry and trigonometry
  • Untimed
  • Endless levels

How it plays

For a small angle measured in radians, sin θ and tan θ are both almost exactly θ. One degree is about 0.01745 radians, so sin 4° ≈ 0.0698 with no table. The rule is worth knowing to the size of its error: sin θ ≈ θ is off by about θ²/6, tan θ ≈ θ by about θ²/3.

Levels go from converting degrees to radians, to sines and tangents, to how far off the rule is at 5° to 20°, to arc against chord, to the height of a tower or the distance of a mast from the angle it fills.

The truth is always the exact trigonometry; a typed answer within 1% counts (5% for the error sizes), and the angles are kept small enough that the small-angle route passes. The wrong options are the real slips: degrees used where radians belong, the sine's error used for the tangent, a factor of ten.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes small-angle estimates and their error sizes quicker and surer. It claims nothing wider.

Learn

How to get better at small angles: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Turn the angle into radians first: multiply the degrees by 0.01745. Every rule here needs the angle in radians; put degrees in and it fails.
  2. Pick the rule the question needs: sin θ ≈ tan θ ≈ θ for values; θ²/6 and θ²/3 for how far off; θ²/24 for arc against chord; size = distance × θ and distance = size ÷ θ for towers and masts.
  3. Size the answer before you choose: the sine of a few degrees is a few hundredths, and an error at 10° is under a few percent.
  4. When typing, give three significant figures. Within 1% counts, or within 5% for the percentage questions.

Techniques and tricks

  • One degree is 0.01745 radians

    Multiply degrees by 0.01745, or by π/180. Learn a few by heart: 10° is 0.1745, 20° is 0.349, 30° is 0.524.

    e.g. sin 4° ≈ 4 × 0.01745 = 0.0698; the true value is 0.06976.

  • One radian is 57.3°

    Going the other way, an angle in degrees divided by 57.3 is radians. This makes distances quick: distance = height × 57.3 ÷ degrees.

    e.g. A 10 m mast at 2°: 10 × 57.3 ÷ 2 = 286.5 m; exactly, 286.4 m.

  • Sine is off by θ²/6, tangent by θ²/3

    sin θ ≈ θ overstates the sine by about θ²/6; tan θ ≈ θ understates the tangent by about θ²/3, twice as much. Square the angle in radians and divide.

    e.g. 10°: 0.1745² = 0.0305; ÷ 6 is 0.51% for the sine, ÷ 3 is 1.02% for the tangent.

  • Double the angle, four times the error

    The error grows with the square of the angle. Know one value and you have the rest.

    e.g. The sine’s error is 0.51% at 10°, so about 2.0% at 20°; the true value is 2.06%.

  • Arc beats chord by θ²/24

    The chord uses the sine of half the angle, which is why the divisor is 24, not 6. Square the whole angle in radians and divide by 24.

    e.g. 30°: 0.524² ÷ 24 = 1.14%; the true value is 1.15%.

  • Size is distance times angle

    For a tower seen from far off, height ≈ distance × angle in radians. It is well within 1% for the angles here.

    e.g. From 1,000 m at 2°: 1,000 × 0.0349 = 34.9 m; exactly 34.92 m.

As the levels climb

  • Level 0 is degrees to radians only. Sines and tangents arrive at level 1, how far off the rule is at level 3, arc against chord and the size of a tower at level 5, and the distance of a mast at level 7. The newest kinds come up more often for a while.
  • As you climb, the angles for sines and tangents widen from up to 4° towards 10° (tangents stop at 8°), and from level 8 the towers stand ten times further off, up to 20,000 m. The rules do not change; the numbers just get longer.
  • The game is untimed unless you set “Time pressure”. With it on, conversions get 20 s, sines 25 s, sizes and distances 30 s, and the how-far-off and arc questions 35 s.

Traps to avoid

  • Degrees where radians belong: sin 4° answered as 4, or a mast’s distance worked as 10 ÷ 2 = 5 m. Convert first, always.
  • A factor of ten: 0.698 for 0.0698. One degree is under two hundredths of a radian, so a few degrees is a few hundredths.
  • The sine’s error used for the tangent: 0.51% for tan 10°, which is off by 1.02%. Tangent is θ²/3, twice the sine’s.
  • The wrong divisor for arc against chord: θ²/6 or θ²/8 instead of θ²/24. It is the half angle inside the chord that makes it 24.
  • cos θ given for sin θ: cos 4° is about 0.998, nearly 1, while sin 4° is nearly 0. A sine of a small angle is small.

How to practise

  • In “Kinds of question”, leave only “How far off” or only “Arc against chord” on to drill the error sizes, which no other kind of question asks for.
  • Set “Answer by” to “Typing the answer”, so you produce three figures rather than picking the one that looks right.
  • Once the rules are quick, turn “Time pressure” on — “Relaxed — 1.5× time” first — and do the arithmetic without paper.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More geometry and trigonometry

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