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Unit circle flashcards: exact values, radians and identities

This starter set covers the unit circle as taught in precalculus and trigonometry: the special angles in degrees and radians, exact sine, cosine and tangent values, quadrant signs, reference angles and the identities you use most. Exact values, the Pythagorean identity and the reciprocal functions were checked against the OpenStax Precalculus text; the double-angle, cofunction and period cards are standard course content. It is a memory set for the circle itself, not a full trigonometry course, so graphing, inverse functions, proofs and equation solving are not included.

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$Flashcard with a unit circle beside a protractor, triangle and open book with quill$

Free Unit Circle flashcards: 33-card starter set

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Degrees and radians (9)

How many radians equal 180 degrees?

pi radians. A full turn, 360 degrees, is 2pi radians.

How do you convert degrees to radians?

Multiply by pi/180. For example, 60 degrees x pi/180 = pi/3.

How do you convert radians to degrees?

Multiply by 180/pi. For example, 3pi/4 x 180/pi = 135 degrees.

What is 30 degrees in radians? What is 45 degrees? 60 degrees?

30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3.

What are 90, 120, 135 and 150 degrees in radians?

90 degrees = pi/2, 120 degrees = 2pi/3, 135 degrees = 3pi/4, 150 degrees = 5pi/6.

What are 210, 225, 240 and 270 degrees in radians?

210 degrees = 7pi/6, 225 degrees = 5pi/4, 240 degrees = 4pi/3, 270 degrees = 3pi/2.

What are 300, 315 and 330 degrees in radians?

300 degrees = 5pi/3, 315 degrees = 7pi/4, 330 degrees = 11pi/6.

What is the definition of one radian?

The central angle of a circle that cuts off an arc equal in length to the radius. In general, angle (radians) = arc length / radius.

What is the arc length formula when the angle theta is in radians?

s = r x theta. It only works in radians, not degrees.

Exact values at the special angles (10)

On the unit circle, what do the x- and y-coordinates of the point at angle theta equal?

x = cos(theta) and y = sin(theta). The point is (cos theta, sin theta).

What are sin(30 degrees) and cos(30 degrees)?

sin(30 degrees) = 1/2 and cos(30 degrees) = √3/2.

What are sin(45 degrees) and cos(45 degrees)?

Both equal √2/2.

What are sin(60 degrees) and cos(60 degrees)?

sin(60 degrees) = √3/2 and cos(60 degrees) = 1/2.

What are sin(0), cos(0), sin(pi/2) and cos(pi/2)?

sin(0) = 0, cos(0) = 1, sin(pi/2) = 1, cos(pi/2) = 0.

What are tan(pi/6), tan(pi/4) and tan(pi/3)?

tan(pi/6) = √3/3 (same as 1/√3), tan(pi/4) = 1, tan(pi/3) = √3.

What is tan(pi/2), and why?

Undefined. tan = sin/cos, and cos(pi/2) = 0, so you would divide by zero.

What are the coordinates of the unit circle point at 5pi/6?

(-√3/2, 1/2). The reference angle is pi/6, and the point is in Quadrant II, so x is negative and y is positive.

What is cos(4pi/3)?

-1/2. The angle is in Quadrant III (cosine negative) with reference angle pi/3, where cosine is 1/2.

What is sin(7pi/4)?

-√2/2. The angle is in Quadrant IV (sine negative) with reference angle pi/4.

Quadrants and reference angles (6)

Which of sine, cosine and tangent are positive in each quadrant?

Quadrant I: all. II: sine only. III: tangent only. IV: cosine only. Memory aid: All Students Take Calculus (A-S-T-C).

What is a reference angle?

The acute angle between the terminal side of an angle and the x-axis. A trig function at the angle has the same size as at its reference angle, with the sign set by the quadrant.

What is the reference angle of 150 degrees?

30 degrees (180 - 150).

What is the reference angle of 7pi/6?

pi/6 (7pi/6 - pi).

What is the reference angle of 315 degrees?

45 degrees (360 - 315).

What is tan(135 degrees)?

-1. The reference angle is 45 degrees, where tangent is 1, and tangent is negative in Quadrant II.

Identities and reciprocal functions (8)

State the Pythagorean identity.

sin^2(theta) + cos^2(theta) = 1. It follows from x^2 + y^2 = 1 on the unit circle.

Define tan, sec, csc and cot using sine and cosine.

tan = sin/cos, sec = 1/cos, csc = 1/sin, cot = cos/sin.

What identity links tangent and secant?

1 + tan^2(theta) = sec^2(theta). Divide sin^2 + cos^2 = 1 by cos^2.

What are sin(-theta) and cos(-theta)?

sin(-theta) = -sin(theta), so sine is odd. cos(-theta) = cos(theta), so cosine is even.

What are the cofunction identities?

sin(90 degrees - theta) = cos(theta) and cos(90 degrees - theta) = sin(theta).

What is the double-angle identity for sine?

sin(2 theta) = 2 sin(theta) cos(theta).

What are the forms of the double-angle identity for cosine?

cos(2 theta) = cos^2(theta) - sin^2(theta) = 2cos^2(theta) - 1 = 1 - 2sin^2(theta).

What is the period of sine and cosine? Of tangent?

Sine and cosine repeat every 2pi radians (360 degrees). Tangent repeats every pi radians (180 degrees).

What this set covers

Topics covered

Degree and radian pairs, exact values of sin, cos and tan, signs by quadrant, reference angles, reciprocal functions, Pythagorean, even-odd, cofunction and double-angle identities

Best used for

Precalculus, trigonometry, and the start of calculus, where exact values are assumed

Card style

Each front names one angle and one function, so there is exactly one correct answer

Not covered

Graphs of trig functions, inverse trig, sum and difference formulas, law of sines, solving trig equations

How to study this set

The unit circle is a pattern you can rebuild, not a table to brute-force. Learn the first-quadrant values and the reflection rules, then use cards to make recall fast enough that it does not cost you time on tests.

1
Day 1
Draw a blank circle and fill in the three first-quadrant angles with their coordinates from memory. Then study the radian conversion cards. Write the answer before you flip.
2
Days 2-7
Add the Quadrant II to IV angles. For each, say the reference angle and the sign out loud before you give the value.
3
Week 2
Add the identity cards. For each identity, derive it once on paper from sin^2 + cos^2 = 1 so you know where it comes from.
4
Week of the test
Shuffle all cards, then redraw the entire circle from scratch with degrees, radians and coordinates. Re-study any angle you hesitate on for more than a few seconds.

Spaced repetition schedules each card to come back just before you would forget it, so reviews get shorter as your recall gets stronger. Cards you miss come back sooner.

Common mistakes to avoid

Swapping sine and cosine

The point at angle theta is (cos theta, sin theta): x first, y second. Students often reverse the pair at 30 and 60 degrees, where the two values are √3/2 and 1/2.

Using the reference angle but forgetting the sign

Reference angles give the size only. Always decide the quadrant and sign separately, for example sin(210 degrees) = -1/2, not +1/2.

Mixing degree and radian modes

Calculators and formulas like s = r x theta need the right unit. Check the mode, and write the unit on every angle.

Treating tan(90 degrees) as 0 or infinity

Tangent is undefined where cosine is zero. At 90 degrees and 270 degrees the value does not exist.

Make your own from your notes

A starter set is a head start, not your whole course. Your class notes, slides and textbook are what your teacher actually tests, so turn them into cards too.

Put one angle and one function on the front, for example "cos(5pi/6)", rather than asking for a whole row of values.
Make separate cards for "convert to radians" and "find the exact value" so you can see which skill is slow.
Add a reference-angle card for each angle you miss, with the quadrant and sign written on the back.
Avoid cards that ask you to recite the whole circle; test single values and rebuild the full circle on paper.

Generate cards from a topic or notes Turn a PDF into flashcards

Unit Circle flashcards: questions

Do I need to memorize the whole unit circle?

Most courses expect fluent recall of the first-quadrant values and the ability to get every other angle from reference angles and signs. You only need to memorize about three angles well plus the pattern for reflecting them.

Why are there two answers for tan(30 degrees)?

There are not two values. √3/3 and 1/√3 are the same number, written with or without a rationalized denominator. Check which form your course or teacher expects.

How do I remember which quadrants are positive?

Use All Students Take Calculus: All are positive in Quadrant I, Sine in II, Tangent in III, Cosine in IV. Then check by thinking of the sign of the x- and y-coordinates.

What is the difference between sin^2(theta) and sin(theta^2)?

sin^2(theta) means (sin theta) squared, so you compute the sine first and then square it. sin(theta^2) squares the angle first. They give different values.

How many unit circle cards should I review per day?

A deck of about 30 cards takes a few minutes. Review the whole set on the days the app schedules, and spend extra time on the angles you answer slowly.

Sources and review

These cards are an independent study aid, not official exam content, and are not affiliated with or endorsed by any exam body or publisher. They were written by hand and checked against the sources below on 2026-10-17. Always confirm details with your course materials and the current official documents.

OpenStax Precalculus 2e: Unit Circle, Sine and Cosine Functions — Checked exact sine and cosine at pi/6, pi/4, pi/3, the Pythagorean identity and the quadrant signs.
OpenStax Precalculus 2e: The Other Trigonometric Functions — Checked tan, sec, csc and cot definitions, tangent values at special angles and reference angle rules.
Wikipedia: Radian — Checked the definition of a radian and degree-radian conversion factors.

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