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Unit Circle Practice

Practice the 16 standard angles of the unit circle five ways: click where an angle sits, name the angle at a marked point, give a point's coordinates, give sin, cos or tan, or solve an equation such as sin θ = 1/2 within one turn. Answers are checked exactly rather than against a rounded decimal, so 1/√2 and √2/2 both count for cos 45°, undefined is the answer for tan 90°, and sin θ = 1/2 accepts both π/6 and 5π/6. Work in degrees, radians or mixed units, where each angle-to-point question also asks for the angle in the other unit. Each answer locks the moment it is given, so a second click cannot change the score, and a set ends with your percentage, a review of every answer and a quick way to practice just the angles you missed. Pick the angles, add a timer or a stopwatch, and print the same set with an answer key.

16 angles, one question each

Angle to point, mixed units

Click the point where each angle sits. In mixed units you also type the angle in the other unit.

Mode
Mixed units
Angles
16 standard
Score
none yet
Correct
none yet
Actions

  • 120°to radians and a point
    2π/3 at (−1/2, √3/2)
  • sin θ = 1/2, within one turntype θ
    π/6 and 5π/6 are both accepted
  • tan 90°an odd multiple of π/2
    undefined, and that is the answer to give

Angles and set number

These take effect when the next set starts. The same set number, angles, units and kind of question always give the same questions in the same order, on screen and on paper.

Choose angles: 16 of 16 selected
Study table: the 16 angles with radians, points and exact values
Accuracy. Every question uses the exact values of the 16 standard angles, written with fractions, √2, √3 and π, and answers are compared exactly: equivalent forms such as 1/√2 and √2/2 both count, while a rounded decimal such as 0.866 does not. Angles are taken within one turn, from 0° up to 360° or 0 up to 2π, and tangent is undefined at 90° and 270°, the odd multiples of π/2. The score is practice feedback, not a grade or a measure of ability.

Common questions

How does this unit circle game work?
Choose degrees, radians or mixed units, which angles to practice, the kind of question and a timer, then start the set. Each chosen angle is asked once, in an order fixed by the set number, so All 16 angles makes a set of 16 questions. Answer by clicking a point on the circle or by typing; on a touch screen, a tap highlights the nearest point and Lock in this point, or a second tap on it, gives it as your answer. Feedback appears as soon as the answer locks: whether it was correct, the exact answer and, for a single angle, its cos, sin and tan. The set ends with your score, your percentage and a review of every answer, and Practice the missed starts a new set on just the angles you missed.
What are the five kinds of question?
Angle to point: click where an angle sits on the circle; in mixed units you also type it in the other unit, so 120° asks for 2π/3 as well. Point to angle: a point is marked with its coordinates, such as (−1/2, √3/2), and you type the angle. Coordinates: type the point (x, y) for an angle. Sin, cos and tan: type the exact value of one of them at an angle. Solve for θ: find θ within one turn where sin, cos or tan of θ equals a value, by clicking a point or typing. All five kinds deals them out evenly and shuffles them.
Which answers count as correct?
Only exact values, compared exactly. Equivalent forms all count: 1/√2, √2/2, sqrt(2)/2 and √(1/2) are the same answer for cos 45°, and 1/√3 and √3/3 both count for tan 30°. A decimal counts only when it is exactly the value, so 0.5 is accepted for 1/2, while 0.866 for sin 60° is marked not correct with a note that it is a rounded decimal. You can type √ or sqrt, and either a hyphen or a minus sign. An answer the page cannot read, such as a sum or a stray letter, is refused with a message saying why and is not scored.
Why does sin θ = 1/2 have two answers?
Because within one turn the sine is 1/2 at two angles, π/6 (30°) and 5π/6 (150°). On screen either one is accepted: click either point, or type one or both separated by commas. An angle outside one turn, such as 13π/6, is refused with a message rather than scored, and 360° or 2π is refused as the same point as 0. The printed set asks for every θ, and its answer key lists both.
Why is tan 90° undefined, and what do I type?
Tangent is sine divided by cosine, and cos 90° is exactly 0, so there is no value to give. Type undefined; undef, DNE and does not exist also count. The same holds at 270°: those are the odd multiples of π/2 within one turn. Typing ∞ or infinity is marked not correct, with a note that the tangent there is undefined rather than infinite.
Does clicking again change my score?
No. Each answer locks as soon as it is given, and a second click or a second typed answer is refused with a note that it does not change the score. In a two-part question, such as a point in mixed units, the click and the typed angle lock separately, and the question counts as correct only when both are right. On a touch screen a tap only highlights the nearest point, so a stray tap costs nothing: tap another point to move the highlight, then press Lock in this point or tap the highlighted point again. The keyboard works the same way as the pointer: Tab to the circle's points, move between them with the arrow keys and press Enter or Space to select one.
Can I practice only radians, or only some angles?
Yes. Choose Degrees, Radians or Mixed units; in mixed units each question is in degrees or radians at random. The angle menu offers all 16 angles, Quadrant I from 0° to 90°, multiples of 30°, multiples of 45° or the 4 axis angles, and Choose angles lets you tick any of the 16. In radians, type angles as multiples of π, such as 5pi/4 or 5π/4; a plain decimal such as 3.927 is read, but it is marked not correct because it is not exact.
Can I print a unit circle worksheet?
Yes. Print a set opens the questions in a new tab, with a small unit circle beside each question that asks you to circle a point or name the angle at a marked one, lines to write on and a separate answer key page, ready to print or save as a PDF. While a set is running it prints that set; otherwise it deals a new one from the settings on screen. Each set has a number, and the same number with the same angles, units and kind of question gives the same questions, so a class can work the printed set and check it on screen. If the browser blocks the new tab, the page says to allow pop-ups.
Is my score kept or graded?
The score is practice feedback for you, not a grade or a measure of ability. The page remembers your settings on this device (units, angles, kind of question and timer) but not your answers or scores. The timer is optional: 30 or 15 seconds a question, where running out counts as not correct, or a stopwatch for the whole set.

Every question uses the exact values of the 16 standard angles, written with fractions, √2, √3 and π, and answers are compared exactly: equivalent forms such as 1/√2 and √2/2 both count, while a rounded decimal such as 0.866 does not. Angles are taken within one turn, from 0° up to 360° or 0 up to 2π, and tangent is undefined at 90° and 270°, the odd multiples of π/2. The score is practice feedback, not a grade or a measure of ability.