117 cards

Unit Circle Flash Cards: Sine, Cosine and Tangent to Flip

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sin 3π/4 (135°)

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Degrees to radians 16

0° in radians 0: no turn at all, so 0° is 0 radians.
30° in radians π/6: multiply by π/180. 30/180 is 1/6 in lowest terms.
45° in radians π/4: multiply by π/180. 45/180 is 1/4 in lowest terms.
60° in radians π/3: multiply by π/180. 60/180 is 1/3 in lowest terms.
90° in radians π/2: multiply by π/180. 90/180 is 1/2 in lowest terms.
120° in radians 2π/3: multiply by π/180. 120/180 is 2/3 in lowest terms.
135° in radians 3π/4: multiply by π/180. 135/180 is 3/4 in lowest terms.
150° in radians 5π/6: multiply by π/180. 150/180 is 5/6 in lowest terms.
180° in radians π: 180° is half a turn, and a full turn is 2π radians.
210° in radians 7π/6: multiply by π/180. 210/180 is 7/6 in lowest terms.
225° in radians 5π/4: multiply by π/180. 225/180 is 5/4 in lowest terms.
240° in radians 4π/3: multiply by π/180. 240/180 is 4/3 in lowest terms.
270° in radians 3π/2: multiply by π/180. 270/180 is 3/2 in lowest terms.
300° in radians 5π/3: multiply by π/180. 300/180 is 5/3 in lowest terms.
315° in radians 7π/4: multiply by π/180. 315/180 is 7/4 in lowest terms.
330° in radians 11π/6: multiply by π/180. 330/180 is 11/6 in lowest terms.

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Radians to degrees 16

0 radians in degrees 0°: 0 radians is no turn at all.
π/2 radians in degrees 90°: put 180° in place of π. 180 ÷ 2 = 90.
2π/3 radians in degrees 120°: put 180° in place of π. 2 × 180 ÷ 3 = 120.
π/3 radians in degrees 60°: put 180° in place of π. 180 ÷ 3 = 60.
3π/2 radians in degrees 270°: put 180° in place of π. 3 × 180 ÷ 2 = 270.
3π/4 radians in degrees 135°: put 180° in place of π. 3 × 180 ÷ 4 = 135.
π/4 radians in degrees 45°: put 180° in place of π. 180 ÷ 4 = 45.
4π/3 radians in degrees 240°: put 180° in place of π. 4 × 180 ÷ 3 = 240.
5π/3 radians in degrees 300°: put 180° in place of π. 5 × 180 ÷ 3 = 300.
5π/4 radians in degrees 225°: put 180° in place of π. 5 × 180 ÷ 4 = 225.
5π/6 radians in degrees 150°: put 180° in place of π. 5 × 180 ÷ 6 = 150.
π/6 radians in degrees 30°: put 180° in place of π. 180 ÷ 6 = 30.
7π/4 radians in degrees 315°: put 180° in place of π. 7 × 180 ÷ 4 = 315.
7π/6 radians in degrees 210°: put 180° in place of π. 7 × 180 ÷ 6 = 210.
11π/6 radians in degrees 330°: put 180° in place of π. 11 × 180 ÷ 6 = 330.
π radians in degrees 180°: π radians is half a turn.

Sine 16

sin 0 (0°) 0: the right end of the circle is the point (1, 0), and sine is its y-coordinate.
sin 2π/3 (120°) √3/2: quadrant II, where sine is positive. The reference angle is π/3 (60°).
sin 3π/2 (270°) −1: the bottom of the circle is the point (0, −1), and sine is its y-coordinate.
sin 3π/4 (135°) √2/2: quadrant II, where sine is positive. The reference angle is π/4 (45°).
sin 4π/3 (240°) −√3/2: quadrant III, where sine is negative. The reference angle is π/3 (60°).
sin 5π/3 (300°) −√3/2: quadrant IV, where sine is negative. The reference angle is π/3 (60°).
sin 5π/4 (225°) −√2/2: quadrant III, where sine is negative. The reference angle is π/4 (45°).
sin 5π/6 (150°) 1/2: quadrant II, where sine is positive. The reference angle is π/6 (30°).
sin 7π/4 (315°) −√2/2: quadrant IV, where sine is negative. The reference angle is π/4 (45°).
sin 7π/6 (210°) −1/2: quadrant III, where sine is negative. The reference angle is π/6 (30°).
sin 11π/6 (330°) −1/2: quadrant IV, where sine is negative. The reference angle is π/6 (30°).
sin π (180°) 0: the left end of the circle is the point (−1, 0), and sine is its y-coordinate.
sin π/2 (90°) 1: the top of the circle is the point (0, 1), and sine is its y-coordinate.
sin π/3 (60°) √3/2: in a 30-60-90 triangle the side across from 60° is √3/2 of the hypotenuse.
sin π/4 (45°) √2/2: a 45-45-90 triangle with hypotenuse 1 has two equal legs of √2/2.
sin π/6 (30°) 1/2: in a 30-60-90 triangle the side across from 30° is half the hypotenuse.

Cosine 16

cos 0 (0°) 1: the right end of the circle is the point (1, 0), and cosine is its x-coordinate.
cos 2π/3 (120°) −1/2: quadrant II, where cosine is negative. The reference angle is π/3 (60°).
cos 3π/2 (270°) 0: the bottom of the circle is the point (0, −1), and cosine is its x-coordinate.
cos 3π/4 (135°) −√2/2: quadrant II, where cosine is negative. The reference angle is π/4 (45°).
cos 4π/3 (240°) −1/2: quadrant III, where cosine is negative. The reference angle is π/3 (60°).
cos 5π/3 (300°) 1/2: quadrant IV, where cosine is positive. The reference angle is π/3 (60°).
cos 5π/4 (225°) −√2/2: quadrant III, where cosine is negative. The reference angle is π/4 (45°).
cos 5π/6 (150°) −√3/2: quadrant II, where cosine is negative. The reference angle is π/6 (30°).
cos 7π/4 (315°) √2/2: quadrant IV, where cosine is positive. The reference angle is π/4 (45°).
cos 7π/6 (210°) −√3/2: quadrant III, where cosine is negative. The reference angle is π/6 (30°).
cos 11π/6 (330°) √3/2: quadrant IV, where cosine is positive. The reference angle is π/6 (30°).
cos π (180°) −1: the left end of the circle is the point (−1, 0), and cosine is its x-coordinate.
cos π/2 (90°) 0: the top of the circle is the point (0, 1), and cosine is its x-coordinate.
cos π/3 (60°) 1/2: in a 30-60-90 triangle the side next to 60° is half the hypotenuse.
cos π/4 (45°) √2/2: at 45° the point is on the line y = x, so cosine and sine are equal.
cos π/6 (30°) √3/2: in a 30-60-90 triangle the side next to 30° is √3/2 of the hypotenuse.

Tangent 16

tan 0 (0°) 0: sine is 0 at the right end of the circle, so sine ÷ cosine is 0.
tan 2π/3 (120°) −√3: quadrant II, where tangent is negative. The reference angle is π/3 (60°).
tan 3π/2 (270°) Undefined: cosine is 0 at the bottom of the circle, and sine ÷ cosine cannot divide by 0.
tan 3π/4 (135°) −1: quadrant II, where tangent is negative. The reference angle is π/4 (45°).
tan 4π/3 (240°) √3: quadrant III, where tangent is positive. The reference angle is π/3 (60°).
tan 5π/3 (300°) −√3: quadrant IV, where tangent is negative. The reference angle is π/3 (60°).
tan 5π/4 (225°) 1: quadrant III, where tangent is positive. The reference angle is π/4 (45°).
tan 5π/6 (150°) −√3/3: quadrant II, where tangent is negative. The reference angle is π/6 (30°).
tan 7π/4 (315°) −1: quadrant IV, where tangent is negative. The reference angle is π/4 (45°).
tan 7π/6 (210°) √3/3: quadrant III, where tangent is positive. The reference angle is π/6 (30°).
tan 11π/6 (330°) −√3/3: quadrant IV, where tangent is negative. The reference angle is π/6 (30°).
tan π (180°) 0: sine is 0 at the left end of the circle, so sine ÷ cosine is 0.
tan π/2 (90°) Undefined: cosine is 0 at the top of the circle, and sine ÷ cosine cannot divide by 0.
tan π/3 (60°) √3: sine ÷ cosine is √3/2 ÷ 1/2, which is √3.
tan π/4 (45°) 1: sine and cosine are equal at 45°, so sine ÷ cosine is 1.
tan π/6 (30°) √3/3: sine ÷ cosine is 1/2 ÷ √3/2 = 1/√3, which is √3/3.

Points on the circle 16

Point at 0 (0°) (1, 0): the right end of the circle. The point is (cosine, sine).
Point at 2π/3 (120°) (−1/2, √3/2): quadrant II. The point is (cosine, sine), so x is negative and y is positive.
Point at 3π/2 (270°) (0, −1): the bottom of the circle. The point is (cosine, sine).
Point at 3π/4 (135°) (−√2/2, √2/2): quadrant II. The point is (cosine, sine), so x is negative and y is positive.
Point at 4π/3 (240°) (−1/2, −√3/2): quadrant III. The point is (cosine, sine), so x is negative and y is negative.
Point at 5π/3 (300°) (1/2, −√3/2): quadrant IV. The point is (cosine, sine), so x is positive and y is negative.
Point at 5π/4 (225°) (−√2/2, −√2/2): quadrant III. The point is (cosine, sine), so x is negative and y is negative.
Point at 5π/6 (150°) (−√3/2, 1/2): quadrant II. The point is (cosine, sine), so x is negative and y is positive.
Point at 7π/4 (315°) (√2/2, −√2/2): quadrant IV. The point is (cosine, sine), so x is positive and y is negative.
Point at 7π/6 (210°) (−√3/2, −1/2): quadrant III. The point is (cosine, sine), so x is negative and y is negative.
Point at 11π/6 (330°) (√3/2, −1/2): quadrant IV. The point is (cosine, sine), so x is positive and y is negative.
Point at π (180°) (−1, 0): the left end of the circle. The point is (cosine, sine).
Point at π/2 (90°) (0, 1): the top of the circle. The point is (cosine, sine).
Point at π/3 (60°) (1/2, √3/2): quadrant I. The point is (cosine, sine), so x is positive and y is positive.
Point at π/4 (45°) (√2/2, √2/2): quadrant I. The point is (cosine, sine), so x is positive and y is positive.
Point at π/6 (30°) (√3/2, 1/2): quadrant I. The point is (cosine, sine), so x is positive and y is positive.

Signs by quadrant 12

Sign of cosine in quadrant I Positive: cosine is the x-coordinate, and quadrant I is right of the y-axis.
Sign of cosine in quadrant II Negative: cosine is the x-coordinate, and quadrant II is left of the y-axis.
Sign of cosine in quadrant III Negative: cosine is the x-coordinate, and quadrant III is left of the y-axis.
Sign of cosine in quadrant IV Positive: cosine is the x-coordinate, and quadrant IV is right of the y-axis.
Sign of sine in quadrant I Positive: sine is the y-coordinate, and quadrant I is above the x-axis.
Sign of sine in quadrant II Positive: sine is the y-coordinate, and quadrant II is above the x-axis.
Sign of sine in quadrant III Negative: sine is the y-coordinate, and quadrant III is below the x-axis.
Sign of sine in quadrant IV Negative: sine is the y-coordinate, and quadrant IV is below the x-axis.
Sign of tangent in quadrant I Positive: tangent is sine ÷ cosine, and in quadrant I they have the same sign.
Sign of tangent in quadrant II Negative: tangent is sine ÷ cosine, and in quadrant II they have opposite signs.
Sign of tangent in quadrant III Positive: tangent is sine ÷ cosine, and in quadrant III they have the same sign.
Sign of tangent in quadrant IV Negative: tangent is sine ÷ cosine, and in quadrant IV they have opposite signs.

Reference angles 9

Reference angle of 2π/3 (120°) π/3 (60°): quadrant II, so 180° − 120° = 60°.
Reference angle of 3π/4 (135°) π/4 (45°): quadrant II, so 180° − 135° = 45°.
Reference angle of 4π/3 (240°) π/3 (60°): quadrant III, so 240° − 180° = 60°.
Reference angle of 5π/3 (300°) π/3 (60°): quadrant IV, so 360° − 300° = 60°.
Reference angle of 5π/4 (225°) π/4 (45°): quadrant III, so 225° − 180° = 45°.
Reference angle of 5π/6 (150°) π/6 (30°): quadrant II, so 180° − 150° = 30°.
Reference angle of 7π/4 (315°) π/4 (45°): quadrant IV, so 360° − 315° = 45°.
Reference angle of 7π/6 (210°) π/6 (30°): quadrant III, so 210° − 180° = 30°.
Reference angle of 11π/6 (330°) π/6 (30°): quadrant IV, so 360° − 330° = 30°.

How this list was compiled: Own work: 117 cards, checked 2026-10-01. Each sine, cosine and tangent is worked out three ways that must agree: by the reference angle and the sign of the quadrant; by turning the point (1, 0) through 15° at a time with the angle-sum formulas in exact arithmetic (fractions and roots, no decimals); and by a computer algebra system. A further check confirms that sine squared plus cosine squared is 1 and that tangent times cosine is sine for every angle, and that each radian measure turns back into its degrees. The same day the finished table was compared with the table of simple algebraic values in https://en.wikipedia.org/wiki/Trigonometric_functions (0°, 30°, 45°, 60° and 90°), and all 16 points were read against the chart of the unit circle in https://en.wikipedia.org/wiki/Unit_circle and the common angles of https://en.wikipedia.org/wiki/Exact_trigonometric_values . Negative answers are stored with the keyboard minus, so a typed -1/2 is marked right and 1/2 is not. Licence: own work

How it works

These unit circle flash cards cover the 16 standard angles from 0 to 11π/6. Each angle is asked six ways: degrees to radians, radians to degrees, sine, cosine, tangent and the point on the circle. The page opens on sine.

Read the full guide

Eight sets under Show only

Show only deals one set. Degrees to radians, Radians to degrees, Sine, Cosine, Tangent and Points on the circle hold 16 cards each; Signs by quadrant has 12 and Reference angles 9. A front gives the angle both ways, as in sin 5π/6 (150°), and the line under each answer says why.

Print a set

Turn a card and the print links appear below it. A set of 16 takes two sheets of Letter or A4, printed on both sides and flipped on the long edge. The backs are mirrored, so 1/2 lands behind sin π/6 (30°).

In Quiz, under More options, type sqrt3/2 for √3/2, pi/6 for π/6 and a hyphen for minus: -1/2.
The check cannot read a root sign, so 3/2 passes for √3/2. Signs are checked: 1/2 is marked wrong for −1/2.
First-quadrant sines run √0/2, √1/2, √2/2, √3/2, √4/2, which is 0, 1/2, √2/2, √3/2, 1. Cosines run the other way.

Questions and answers

What are the 16 angles on the unit circle?

0°, 30°, 45°, 60° and 90°, then the same reference angles in the other three quadrants: 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315° and 330°. In radians they run from 0 to 11π/6.

How do you memorize the unit circle?

Start with the first quadrant: the points at 30°, 45° and 60° are (√3/2, 1/2), (√2/2, √2/2) and (1/2, √3/2). Every other point reuses those numbers, and the quadrant sets the signs.

Why is tan π/2 undefined?

Tangent is sine divided by cosine. At π/2 (90°) the point on the circle is (0, 1), so cosine is 0, and nothing can be divided by 0. The same happens at 3π/2 (270°).

How do you convert degrees to radians?

Multiply by π/180 and reduce the fraction: 150° is 150π/180, which is 5π/6. To go back, put 180° in place of π: 5 × 180 ÷ 6 = 150°.