How to memorize the unit circle (it's really only three numbers)

The whole unit circle

The unit circle has a radius of 1, so the point where each angle meets it is (cos θ, sin θ). These are the 16 standard angles:

That looks like 48 things to memorize. It’s really three patterns.

Pattern 1: the first quadrant is a square-root count

In the first quadrant, the sines of 0°, 30°, 45°, 60° and 90° count up under a square root:

sin: √0/2, √1/2, √2/2, √3/2, √4/2

which simplifies to 0, 1/2, √2/2, √3/2, 1. The cosines are the same list backwards: 1, √3/2, √2/2, 1/2, 0. So in the whole first quadrant there are only three numbers to know: 1/2, √2/2 and √3/2.

A quick check for 30° versus 60°: a small angle gives a small height, so sin 30° is the small one, 1/2.

The hand trick, which math departments such as the University of Houston’s hand out, does the same count on your fingers:

1. Number your fingers 0°, 30°, 45°, 60° and 90° from pinky to thumb.
2. Fold down the finger for your angle.
3. The fingers on the pinky side give the sine and the fingers on the thumb side give the cosine, each as √(count)/2.

Example: For 60°, fold your index finger. Three fingers on the pinky side make sin 60° = √3/2, and one on the thumb side makes cos 60° = √1/2 = 1/2.

Pattern 2: reference angles copy the first quadrant everywhere

Every angle outside the first quadrant has a reference angle, the acute angle it makes with the x-axis, and it has the same cosine and sine as that reference angle, apart from the signs.

• 150° is 30° short of 180°, so it uses the 30° values: (−√3/2, 1/2).
• 225° is 45° past 180°, so it uses the 45° values: (−√2/2, −√2/2).
• 300° is 60° short of 360°, so it uses the 60° values: (1/2, −√3/2).

Tip: In radians, the denominator tells you the family: anything over 6 is a 30° angle, over 4 a 45° angle, over 3 a 60° angle.

Pattern 3: the signs come from the quadrant

Remember All Students Take Calculus (OpenStax’s textbook uses “A Smart Trig Class”), starting in the top right and going counterclockwise:

It’s just the coordinate plane: cosine is x, so it’s negative on the left; sine is y, so it’s negative below the axis.

The mix-ups that cost points

• Swapping cos and sin. The point is (cos θ, sin θ): x first, then y, like c before s in the alphabet.
• Swapping 30° and 60°. sin 30° = 1/2 and cos 30° = √3/2; at 60° they trade places.
• Wrong sign in quadrant II, III or IV. Find the quadrant first, then the values.
• π/6 versus π/3. A bigger denominator means a smaller angle: π/6 is 30° and π/3 is 60°.
• Tangent. tan θ = sin θ / cos θ, so tan 45° = 1 and tan 90° is undefined, because cos 90° = 0.

A week that makes it stick

Spreading practice over several days is one of only two study techniques rated high utility in a major review (Dunlosky et al., 2013), and the longer you need to remember something, the wider your review gaps should be (Cepeda et al., 2006).

More lists worth memorizing

The same approach works for any fixed list your class tests: group it, find the pattern, then let spaced flashcards do the rest. See how to memorize the 12 cranial nerves, the 20 amino acids the polyatomic ions and the periodic table, each with a free deck.