How do you memorize the Unit Circle?

Question

The unit circle can be daunting. It is necessary in all higher math classes and I am positive you will have a quiz or test on filling in the X and Y coordinates for each of the angles in the Unit Circle. I have an easy way to ace your quiz or test in filling in the Unit Circle.

Answer 1

There is an easy way to memorize the unit circle. If you ask me, I never actually memorized the unit circle. I learned it using the “hand trick” method from a YouTube video. You can search it up. It shows you how to find the sine and cosine of the first quadrant. Once you learn the first quadrant, you will realize the pattern repeats throughout the entire unit circle.

The pattern goes as follows: for the first quadrant, the main angles are 0°, 30°, 45°, 60°, and 90°. The x-coordinate is cosine and the y-coordinate is sine. The cosine values go from biggest to smallest: 1, √3/2, √2/2, 1/2, and 0. The sine values go from smallest to biggest: 0, 1/2, √2/2, √3/2, and 1. So, the coordinates are:

0° = (1, 0)

30° = (√3/2, 1/2)

45° = (√2/2, √2/2)

60° = (1/2, √3/2)

90° = (0, 1)

After you know the first quadrant, the rest of the unit circle becomes much easier because the same coordinate values repeat. The only thing that changes is whether the x-value or y-value is positive or negative. In Quadrant I, both x and y are positive. In Quadrant II, x is negative and y is positive. In Quadrant III, both x and y are negative. In Quadrant IV, x is positive and y is negative.

A helpful way to remember this is: cosine is the x-coordinate and sine is the y-coordinate. We also know that in the first quadrant, both the x-coordinate and y-coordinate are positive. For example, the point (3, 4) would be in the first quadrant because both 3 and 4 are positive. Since cosine represents the x-coordinate and sine represents the y-coordinate, that means both cosine and sine are positive in the first quadrant.

You can also remember the signs using the mnemonic “All Students Take Calculus.” “All” represents the first quadrant, which means all trig functions are positive there. “Students” represents the second quadrant, which means only sine is positive. “Take” represents the third quadrant, which means only tangent is positive. “Calculus” represents the fourth quadrant, which means only cosine is positive. Once you understand this idea, it becomes much easier to figure out the signs in each quadrant instead of memorizing every single value separately.

Answer 2

I use the following "unforgettable chart" from my high school trig teacher ("Boston" Bob Connolly):

n 0 1 2 3 4 n is the column which numbers the angles in Q1

θ 0 π/6 π/4 π/3 π/2 θ is the angle in radians in Q1

sinθ=√n/2 0 1/2 √2/2 √3/2 1 √n/2 generates the exact values of sinθ

cosθ 1 √3/2 √2/2 1/2 0 cosθ is sinθ backwards (the complement)

tanθ 0 1/√3 1 √3 ∞ tanθ = sinθ/cosθ

The key is plugging in each column number n in sequence, noting for n=4 that √4/2 = 2/2 = 1.

Just remember the formula √n/2 and you can construct sinθ and cosθ whenever you forget them.

That gets you Quadrant 1 and you get sinθ and cosθ in the other quadrants by using the appropriate sign.

Answer 3

Know SOHCAHTOA, know your 30-60-90 triangles, 45-45-90 degree triangles; know that there are 2 pi radians in a circle, and thus what pi/2, pi/3, pi/4 and pi/6 mean, draw little triangles on a coordinate plane with reference angles on the x- axis then you can construct that circle that students are often asked to memorize. But there is a reason you are being asked to memorize it, THE INFORMATION THAT IT CONTAINS WILL BE NEED AS YOU PROCEED IN HIGHER MATH. Draw pictures of those triangles, understand when certain triangles drawn on the plane generate positive and negative trig ratios and be confident that you can figure sin, cos, and tan of any of 30-45-60-90 reference angles in any quadrant and you'll be good.