How do I remember the angles in the unit circle?

The unit circle is a circle of radius one. For each angle, there is a point on the circle whose x-coordinate is the cosine value of the angle and whose y-coordinate is the sine value of the angle. There are patterns within the unit circle that make it easier to understand and to memorize.

ANGLES (in radians):

First, the radian angle measures of the four “corner” points on the circle are 0, pi/2, pi, 3pi/2, and 2pi. The 2pi angle is one complete full circle around the unit circle and is in the same position as the 0 angle measure.

Next, it’s easiest to identify the “pi/4” angles, as they are each in the exact middle of a quadrant. Pi/4 can be marked in the middle of the first quadrant (Quadrant I), 3pi/4 is in the middle of the second quadrant, 5pi/4 is in the middle of the third quadrant, and 7pi/4 is in the middle of the fourth quadrant.

Next, the “pi/6” angles are positioned closer to the x-axis and are pi/6, 5pi/6, 7pi/6, and 11pi/6.

Finally, the “pi/3” angels are positioned closer to the y-axis and are pi/3, 2pi/3, 4pi/3, and 5pi/3.

(X,Y) POINT FOR EACH ANGLE GIVES YOU SIN AND COS:

There are patterns to remember the (x,y) coordinates of the point on the circle that corresponds to each angle mentioned above.

Since each of the four “corner” points at 0, pi/2, 3pi/2, and 2pi is a distance of one full unit from the origin center of the circle, their coordinates are each (1,0), (0,1), (-1,0), and (0, -1), respectively.

NOTE: For the other angles, you only need to remember these three important numbers:

1/2

(square-root of 2)/2

(square-root of 3)/2

You just need to remember that:

1) The SMALLEST of these numbers is 1/2

2) The MID-SIZED number is (square-root of 2)/2

3) The LARGEST of these numbers is (square-root of 3)/2

For each of the remaining angles (for instance pi/6, pi/4, pi/3, etc), if the corresponding point on the circle has the smallest possible x-distance, its x-coordinate is 1/2, and if it has the largest possible x-distance, its x-coordinate is (square-root of 3)/2. If it has the middle distance, its coordinate value is (square-root of 2)/2. The same is true for the y-values.

NOTE: Remember that the x-coordinate is the cos value, and the y-coordinate will give you the sin value.