How do you memorize the unit circle?
Memorize only the first quadrant, then use patterns for the rest. Learn three points: 30° is (√3/2, 1/2), 45° is (√2/2, √2/2) and 60° is (1/2, √3/2). Every other angle reuses these numbers with different signs. Use ASTC for the signs and a simple numerator rule for the radians, then quiz yourself until you can fill in a blank circle in a few minutes.
The unit circle is a circle with radius 1 around the origin. For any angle, the point on the circle is (cos θ, sin θ). So memorizing the circle means knowing cosine and sine for 16 standard angles. That sounds like 32 values, but there are only three numbers to learn, 1/2, √2/2 and √3/2, plus 0 and 1.
In the first quadrant, the x values count down and the y values count up. Put everything over 2: x goes √3, √2, √1 and y goes √1, √2, √3 as the angle goes 30°, 45°, 60°. Since √1 is 1, that gives the three points above. Tangent is sine divided by cosine: √3/3, 1 and √3.
The other quadrants mirror the first. The reference angle, the angle back to the x-axis, gives the numbers, and the quadrant gives the signs. A common way to remember the signs is ASTC, All Students Take Calculus: all three functions are positive in quadrant I, only sine in II, only tangent in III and only cosine in IV.
| Degrees | Radians | (cos, sin) | tan |
|---|---|---|---|
| 0° | 0 | (1, 0) | 0 |
| 30° | π/6 | (√3/2, 1/2) | √3/3 |
| 45° | π/4 | (√2/2, √2/2) | 1 |
| 60° | π/3 | (1/2, √3/2) | √3 |
| 90° | π/2 | (0, 1) | undefined |
- 1Learn the three first-quadrant points with the count-down, count-up rule: x is √3, √2, √1 over 2 and y is √1, √2, √3 over 2.
- 2Keep the hand trick as a backup. Hold up your left hand, palm facing you, and label the pinky 0°, then 30°, 45°, 60° and the thumb 90°. Fold the finger for your angle. The square root of the fingers on the thumb side, over 2, is the cosine. The square root of the fingers on the pinky side, over 2, is the sine.
- 3Fill in the degrees for all four quadrants with the pattern +30, +15, +15, +30: 30, 45, 60, 90, then 120, 135, 150, 180, and so on up to 360.
- 4Get the radians from the denominators 6, 4 and 3. In quadrant II the numerator is the denominator minus 1 (5π/6, 3π/4, 2π/3). In quadrant III it is the denominator plus 1 (7π/6, 5π/4, 4π/3). In quadrant IV it is twice the denominator minus 1 (11π/6, 7π/4, 5π/3).
- 5Copy the first-quadrant numbers into each quadrant and set the signs with ASTC.
- 6Draw a blank circle from memory once a day for a week and check it. Then turn single points into flashcards, such as sin 5π/6 on the front and 1/2 on the back, and review them spaced out over the next weeks.