absurdtools

Unit Circle Trainer

Drill the exact trig values — √3/2, not 0.866 — with the whole circle held exactly, because a computer cannot check it by evaluating.

The whole circle

Degrees Radians sin cos tan what the machine says for sin how far out
0° 0 0 1 0 0 exact
30° π/6 1/2 √3/2 √3/3 0.49999999999999994 5.55e-17
45° π/4 √2/2 √2/2 1 0.7071067811865475 1.11e-16
60° π/3 √3/2 1/2 √3 0.8660254037844386 exact
90° π/2 1 0 undefined 1 exact
120° 2π/3 √3/2 -1/2 -√3 0.8660254037844387 1.11e-16
135° 3π/4 √2/2 -√2/2 -1 0.7071067811865476 exact
150° 5π/6 1/2 -√3/2 -√3/3 0.49999999999999994 5.55e-17
180° π 0 -1 0 1.2246467991473532e- 1.22e-16
210° 7π/6 -1/2 -√3/2 √3/3 -0.5000000000000001 1.11e-16
225° 5π/4 -√2/2 -√2/2 1 -0.7071067811865475 1.11e-16
240° 4π/3 -√3/2 -1/2 √3 -0.8660254037844385 1.11e-16
270° 3π/2 -1 0 undefined -1 exact
300° 5π/3 -√3/2 1/2 -√3 -0.8660254037844386 exact
315° 7π/4 -√2/2 √2/2 -1 -0.7071067811865477 1.11e-16
330° 11π/6 -1/2 √3/2 -√3/3 -0.5000000000000004 4.44e-16

The last two columns are what JavaScript returns and how far that is from the truth. At 180° the machine gives 1.2246467991473532e-16 where the answer is 0, because the nearest double to π is not π. Across every value on this circle it is exactly right 13 times out of 46, and its worst miss is 1.78e-15, on tan 240°. Small enough never to matter in a drawing, and quite enough to mean the machine cannot be the thing you check your answers against.

Why a computer cannot check this by evaluating

  • A computer cannot check the unit circle by evaluating it. sin(π) is 0, but ask JavaScript and it answers 1.2246467991473532e-16. tan(π/2) is undefined, and it answers 16331239353195370. Neither is a bug: π is irrational, the nearest double to π is not π, and every trig value a machine gives at these angles is the value at a slightly wrong angle.
  • So every value here is stored exactly, as a signed square root over a denominator, and answers are checked against that rather than against a decimal. Marking √3/2 as merely close to 0.8660254 would defeat the point of learning it.
  • The first quadrant is worth seeing as one pattern rather than five facts. Sine at 0, 30, 45, 60 and 90 degrees is √0/2, √1/2, √2/2, √3/2, √4/2 — and cosine is the same list read backwards. That is why these values are memorable at all.
  • Tangent is undefined at 90° and 270°, which is a different statement from being very large. Cosine is exactly zero there, and dividing by zero does not give infinity — it gives nothing. The drill accepts "undefined" and nothing else.
  • Decimals are refused on purpose. 0.866 is not sin 60°, it is a rounded approximation to it, and a trainer that accepts one teaches that the two are interchangeable when the entire exercise is that they are not.

How to use

  1. Pick which topics you want to practise.
  2. Answer with exact values — decimals are marked wrong on purpose.
  3. Check the full table underneath, machine values included.

Frequently asked questions

Why is 0.866 marked wrong for sin 60°?

Because it is not the answer. sin 60° is exactly √3/2, and 0.866 is a rounded approximation to it. Accepting one for the other teaches that they are interchangeable, when the entire exercise is that they are not — so decimals are refused on purpose.

Why not just check answers by evaluating them?

Because a computer cannot. Ask JavaScript for sin(π) and it gives 1.2246467991473532e-16, not 0. Ask for tan(π/2) and it gives 16331239353195370, not "undefined". π is irrational, the nearest double to π is not π, and every trig value a machine returns at these angles is the value at a slightly wrong angle.

So how are the values stored?

Exactly, as a signed square root over a denominator, and answers are compared against that. All 16 angles were checked to satisfy the identities that define them — sine squared plus cosine squared is 1 everywhere, and tangent equals sine over cosine everywhere it is defined.

Is there a pattern worth learning?

Yes, and it is the reason these values are memorable. Sine at 0, 30, 45, 60 and 90 degrees is √0/2, √1/2, √2/2, √3/2, √4/2 — and cosine is the same list read backwards. Five separate facts become one pattern.

Why is tan 90° "undefined" rather than infinity?

Because cosine is exactly zero there, and dividing by zero does not give infinity — it gives nothing at all. The drill accepts "undefined" and refuses both "infinity" and the enormous number a calculator would show, since those describe a limit rather than a value.

Does the table show what my computer says?

It does, in the last column, so you can compare. At 180° the exact sine is 0 and the machine returns 1.2246467991473532e-16. Twenty-two of the thirty-two sine and cosine values differ from the exact ones — slightly, but not by nothing.

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