Unit Circle Practice

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.

How can I use the game to memorize the unit circle?

Start with Quadrant I and Timer off. Use Angle to point to match each angle to its position, then Point to angle to recall the angle from a marked point. Check the Study table when you need the exact values. After a set, use Practice the missed to repeat the angles you got wrong; switch to All 16 angles when you want a full review.

Can I play a fill-in-the-blank unit circle game?

Yes. Choose Point to angle to fill in the angle, Coordinates to fill in (x, y), or Sin, cos and tan to fill in an exact value. Each question has a typed answer box and a Check button. The game checks one question at a time; it does not ask you to label every point on a blank circle at once.

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. Write a product as one exact value: multiplication signs between numbers are refused without scoring. For radian angles, a coefficient times π such as 5*pi/4 is accepted. 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.

Can I print a unit circle board game?

Yes. Print board game opens an original 16-space board, the seeded questions and a separate answer key. Two players supply physical tokens and take turns answering the next question. The other player checks the key: a correct answer moves the token one numbered space, and a wrong answer leaves it in place. Give every requested part, including all angles in a printed solve question. After the last question, start again at question 1; the first token to space 16 wins. While a set is running, the board uses that set; otherwise it uses the settings on screen. Print or save as PDF in the new tab.

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.