Memorize the Unit Circle

Learning the unit circle will help you not only in trigonometry and geometry, but in future calculus courses as well. It may look like a lot to memorize, but once you understand how it works, you can start with a few numbers and quickly figure out the rest.

Steps

Memorizing Radians

1. Draw two perpendicular lines. Place a straightedge on a large sheet of paper. Draw one vertical and one horizontal line. These should intersect near the center of the page. These are the x and y axes of a graph.
2. Draw-a-Circle. Using a compass, draw a large circle with its center at the intersection of the two lines.
3. Understand radians. A radian is a measure of angle. Specifically, it is defined such that a person walking around a circle with a radius of 1 unit sweeps across an angle of one radian after walking for 1 unit around the circumference. In the next step, we'll label the four coordinate points with the radian value. If you memorize the formula to relate circumference and radius, you can quickly work these out from scratch, even if you haven't memorized them.
   • The unit circle's radian measurements always assume you are starting from the (0, 1) coordinate. To make it clear which point we're referring to, we'll describe the circle as a compass rose:
4. Remember that the circle's circumference is 2π. A circle's circumference equals 2πr, where r is the radius. Since the unit circle has a radius of 1, we can simplify its circumference to 2π. You can find the radian value of any point on the circumference, just by taking 2π and dividing by the fraction of the circle you've covered. This is much easier than memorizing every single value on the circle.
5. Label the four points on the x and y axes. All this takes is dividing 2π into quarters:
   • "East" is the starting point, so you've covered 0 radians.
   • "North" = ¼ of the circumference = ^2π/₄ = ^π/₂ radians.
   • "West" = halfway across = ^2π/₂ = π radians.
   • South = three quarters across = 2π * ¾ = ^3π/₂ radians.
   • Walking the entire circumference brings you back to your starting point. You can label this 2π as well as 0 to represent this.
6. Divide the circle into eighths. Now draw a line through each quadrant, cutting it perfectly in half. Once again, use division to find the value in radians:
   • ^π/₄
   • ^3π/₄
   • ^5π/₄
   • ^7π/₄
   • (π/2, π, 3π/2, and 2π are already labeled.)
7. Divide the circle into six segments. Now draw additional lines cutting the circle into six segments. (You can use a protractor to do this, starting at the positive x-axis and measuring 60 degrees each time.) You can use the same approach above to realize that one-sixth of a circle covers ^2π/₆ = ^π/₃ radians. Use this to label the following points on the circumference (one in each quadrant):
   • ^π/₃
   • ^2π/₃
   • ^4π/₃
   • ^5π/₃
   • (π and 2π are already labeled)
8. Draw in the twelfths. The last points labeled on most unit circles represent increments of one-twelfth the circumference. Only four of these have not yet been labeled:
   • ^π/₆
   • ^5π/₆
   • ^7π/₆
   • ^11π/₆

Memorizing X-Y Coordinates (Cosine, Sine)

1. Understand cosine and sine. The unit circle is especially useful for right angle trigonometry. Each x coordinate of a point on the circle equals cosine(θ), and each y coordinate equals sine(θ), where θ is the value of the angle.
   • If you have trouble remembering this, think (cos, sin) "'cause sine goes last".
   • You can derive this using right triangles and the definition of these functions — remember "sohcahtoa"?
2. Write the coordinates on four points of the circle. A "unit circle" is just a circle with a radius exactly one unit long. Use this to find the x and y coordinates of four points on the circle, where it intersects with an axis. (We're calling this "East," "North," and so on for ease of reading, but these are not official names.)
   • The "East" point's coordinates are (1, 0).
   • The "North" point is at (0, 1).
   • The "West" point is at (-1, 0).
   • The "South" point is at (0, -1).
   • This works just like a normal graph. You should be able to figure out these coordinates on your own, without having to memorize them.
3. Memorize coordinates for the first quadrant. The first quadrant is the upper right quarter of the circle, where both x and y values are positive. These are the only coordinate values you have to memorize:
   • At ^π/₆, the coordinates are (<math>\frac{\sqrt{3}}{2},\frac{1}{2}</math>).
   • At ^π/₄, the coordinates are (<math>\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}</math>).
   • At ^π/₃, the coordinates are (<math>\frac{1}{2},\frac{\sqrt{3}}{2}</math>).
   • Notice that there are only three numerators. Moving in a positive direction (left to right for x values, bottom to top for y values), these should go 1 → √2 → √3.
4. Draw straight lines to fill in other coordinates. If you can draw a perfectly vertical or a perfectly horizontal line between two points, they have the same absolute value x- and y- coordinates. In other words, you can draw a line from a point in the first quadrants, write in the same coordinates at the point where you end up, and leave room to right in the sign (+ or -).
   • For example, you can draw a horizontal line between ^π/₃ and ^2π/₃. Since the coordinates at the first point are (<math>\frac{1}{2},\frac{\sqrt{3}}{2}</math>), the coordinates at the second point are (?<math>\frac{1}{2},?\frac{\sqrt{3}}{2}</math>), where "?" is standing in for a + or - sign.
   • Here's a shortcut: check the denominator in the radians. All points ending /3 have the same absolute value coordinates, as do all points ending in /4 and all points ending in /6.
5. Use symmetry to figure out if it's positive or negative. There are several ways to memorize where to put the - signs on your circle:
   • Think of basic graph rules. Above the x axis is positive, below is negative. Left of the y axis is negative, right is positive.
   • Start from quadrant 1 and draw lines to other points. If the line crosses the y axis, the y value switches sign. If it crosses the x axis, the x value switches sign.
   • Memorize "All Students Take Calculus" (ASTC), moving counter-clockwise. Quadrant 1 has All positive values, Q2 has positive Sine only, Q3 has positive Tangent only, and Q4 has positive Cosine only.
   • Whichever method you choose, the signs are (+,+) for quadrant 1, (-,+) for quadrant 2, (-,-) for quadrant 3, and (+,-) for quadrant 4.
6. Check your work. Here's the full list of coordinate values at every labeled point on your circle (not counting the four points on the axes), moving clockwise. Remember, you should be able to find all these values just by memorizing quadrant 1:
   • Quadrant 1: (<math>\frac{\sqrt{3}}{2},\frac{1}{2}</math>); (<math>\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}</math>); (<math>\frac{1}{2},\frac{\sqrt{3}}{2}</math>).
   • Quadrant 2: (<math>-\frac{1}{2},\frac{\sqrt{3}}{2}</math>); (<math>-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}</math>); (<math>-\frac{\sqrt{3}}{2},\frac{1}{2}</math>)
   • Quadrant 3: (<math>-\frac{\sqrt{3}}{2},-\frac{1}{2}</math>); (<math>-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}</math>); (<math>-\frac{1}{2},-\frac{\sqrt{3}}{2}</math>)
   • Quadrant 4: (<math>\frac{1}{2},-\frac{\sqrt{3}}{2}</math>); (<math>\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}</math>); (<math>\frac{\sqrt{3}}{2},-\frac{1}{2}</math>)

Tips

• The process will shorten considerably with practice. In the future, you may only need to see the x- and y-axis, or you may not even need a diagram.
• If you have a quiz or test focused on the unit circle, sketch the circle on your scratch paper first so you can refer to it for every problem.

Related Articles

• Understand the Unit Circle

See also

• Use Right Angled Trigonometry
• Remember the Trigonometric Table
• Use the Sine Rule
• Learn Trigonometry
• Find the Perimeter of a Triangle
• Use the Cosine Rule
• Convert Degrees to Radians
• Solve Trigonometric Equations
• Solve Trigonometric Inequalities
• Understand the Unit Circle

What links here

• Understand the Unit Circle