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Unit circle

sin 150°, cos 5π/4, tan 300°: exact values from the two special triangles and the sign of the quadrant, in degrees and radians.

  • Geometry and trigonometry
  • Timed
  • Endless levels

How it plays

An angle that is a multiple of 30° or 45° and a function of it; pick the exact value — 1/2, √2/2, −√3, not defined. It starts with sin between 0° and 90°; cos, tan, quadrant signs and sec, csc, cot unlock as you climb, then radians, and the range keeps widening: a full turn, then negative angles, two turns, three, without end.

The wrong options are the usual slips: the sign the quadrant gives forgotten, sin taken for cos, the reciprocal, the other acute angle of the triangle.

The working derives every answer: bring the angle into one turn, find its quadrant and reference angle, read the ratio off the 30-60-90 triangle (sides 1, √3, 2) or the 45-45-90 one (1, 1, √2), then give it the quadrant's sign.

Keys

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

What it trains

This makes the exact trigonometric values of the special angles, and their signs, quick to produce. It trains that, nothing wider.

Learn

How to get better at unit circle: 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. If the angle is in radians, turn it into degrees: π is 180°, so 5π/6 is 5 × 30° = 150°.
  2. Bring it into one turn: add or take away 360° until it lies from 0° to 360°. −30° is 330°; 675° is 315°.
  3. Find its quadrant and its reference angle — the angle back to the nearest side of the x-axis: 180° − θ in II, θ − 180° in III, 360° − θ in IV.
  4. Read the value for the reference angle off the special triangle, then give it the quadrant’s sign: sin takes the sign of y, cos of x, tan is positive where x and y share a sign.

Techniques and tricks

  • Two triangles, no table

    Half an equilateral triangle of side 2 has sides 1, √3, 2 and angles 30°, 60°, 90°. Half a unit square has sides 1, 1, √2. sin is opposite ÷ hypotenuse, cos adjacent ÷ hypotenuse, tan opposite ÷ adjacent.

    e.g. tan 60° = √3 ÷ 1 = √3; cos 30° = √3 ÷ 2.

  • The square-root count

    For 0°, 30°, 45°, 60°, 90°, sin is √0/2, √1/2, √2/2, √3/2, √4/2: that is 0, 1/2, √2/2, √3/2, 1. cos is the same list read backwards.

  • All Students Take Calculus

    Going round from quadrant I: All three are positive in I, only Sin in II, only Tan in III, only Cos in IV. csc shares sin’s sign, sec shares cos’s, cot shares tan’s.

    e.g. cos 120°: quadrant II, so negative — −1/2.

  • Radians in sixths and quarters

    π/6 is 30°, π/4 is 45°, π/3 is 60°. So kπ/6 is k × 30° and kπ/4 is k × 45°.

    e.g. 11π/6 = 330°; 5π/4 = 225°; −10π/3 = −600°, which is 120° after two full turns.

  • On an axis, read the point

    At 0°, 90°, 180° and 270° the point on the circle is (1, 0), (0, 1), (−1, 0) and (0, −1). sin is y, cos is x, tan is y ÷ x — not defined where x is 0, at 90° and 270°.

  • Expect tidy surds

    Answers are written with the root on top: tan 30° is √3/3, not 1/√3, and sec 30° is 2√3/3, not 2/√3. Multiply top and bottom by the root to match.

As the levels climb

  • It starts with sin from 0° to 90°. cos arrives at level 1, tan at level 5, the quadrant and sign questions at level 8, and sec, csc and cot at level 11.
  • The range of angles widens: to 180° at level 2, a full turn at level 6, then ±360° at level 9, ±720° at level 12, and another turn each level after that, without end. Past one turn, add or take away whole turns first.
  • With “Angles in” left on “Degrees, then radians as you climb”, radians are mixed in from level 3. They get 3 s more, and negative angles or angles past a full turn 2 s more.
  • The clock tightens to 80% at level 4, 65% at level 7 and 55% at level 10, from 6 s at the start for a value. By then the reference angle and sign should come before you finish reading.

Traps to avoid

  • The sign forgotten: cos 120° answered 1/2. Find the quadrant before the value, so the sign is never an afterthought.
  • sin taken for cos: cos 30° answered 1/2. Adjacent ÷ hypotenuse at 30° is the long leg, √3, over 2.
  • The reciprocal: sec 60° answered 1/2, which is cos 60°. sec is 1 ÷ cos, so 2.
  • The other acute angle: sin 60° answered 1/2, which is sin 30°. The side opposite 60° is the long one, √3.
  • tan 90° answered 0 or 1. At 90° the point is (0, 1), and y ÷ x divides by zero: not defined.

How to practise

  • In “Questions”, leave only “sec, csc, cot” on for a while, so the reciprocals come as fast as sin and cos.
  • Set “Angles in” to “Radians only” once degrees are quick: the conversion has to become as fast as the value.
  • Raise “Options shown” to 8 so every classic slip — the wrong sign, the partner, the reciprocal — sits beside the truth at once.

Levels

Each question has a clock that comes from the work it asks for. 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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