# How to Remember the Unit Circle (NancyPi)

For my video on TRIGONOMETRY basics and trigonometric functions (sin, cos, tan, csc, sec, cot trig functions and basic trigonometry), jump to: https://youtu.be/bSM7RNSbWhM

Unit circle trigonometry comes up a lot in geometry, precalculus, and even calculus problems. The trig unit circle is a circle of radius one. For each angle, there is a point on the trig circle whose x-coordinate is the cosine value of the angle and whose y-coordinate is the sine value of the angle. It is a way of finding exact values of trig functions. Here is how to memorize the unit circle values, how to fill in the unit circle, and how to use the unit circle:

First, the radian angle measures of the four “corner” points on the radian circle are 0, pi/2, pi, 3pi/2, and 2pi. The 2pi angle is one complete full circle around the unit circle (radians) 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 trig values:

There are patterns to remember the (x,y) coordinates of the point on the trigonometric circle (“pi 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 unit circle 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: When doing unit circle practice, working on a unit circle worksheet, or studying for a unit circle quiz, remember that the x-coordinate is the COS value, and the y-coordinate will give you the SIN value. For instance, for the unit circle point (1,0) at angle 0, the value of cos 0 is the x-value 1, and the the value of sin 0 is the y-value 0.

For how to find TAN (tangent) from the sin and cos values, jump to my trig video at https://youtu.be/bSM7RNSbWhM