How to Memorize the Unit Circle?

Students who opted for Mathematics as a subject must have come across the topic ‘unit circle’. It is a simple tool for assistance in solving complex mathematical problems. Regular topic revision is necessary to keep it fresh in mind and apply it instantly as required. But, to carry out the memorization process, there is no need to cram up the unit circle. Rather, you can simply understand how it is created and practice it a few times to set it in your mind. 

Wondering how to understand it? Let’s begin!

What is a Unit Circle? 

As unit means ‘single’ or ‘one,’ a unit circle is a circle with a radius of 1 and a center at (0,0). A unit circle is a significant tool for finding the values of trigonometric functions and angle measurements by simply understanding the circle. 

Alt text: Diagrammatic Representation of Unit Circle

Figure Source: Remind

Methods to Memorize the Unit Circle

There are different methods to help you memorize the unit circle. Let’s understand each of them. 

Method 1: For Angle in Radians

Finding the x-axis and y-axis

Remember that the x-axis is not a fraction, while the y-axis itself is a fraction. The positive side of the x-axis is 2 while the opposing side is simply . Moreover, the positive y-axis is 2, and the negative y-axis is 32. 

Finding the Numerator of Angle in Radians

You need to remember the combination of letters in the following specific format: ASAP. You can remember it as mnemonics or as a mentioned acronym. Since the values on each part of the circle will be different, you need to follow the full form of ‘ASAP,’ which is ‘A’ for All, ‘S’ for Subtract, ‘A’ for Add and ‘P’ for Prime numbers. 

Let’s learn how you will use it. 

‘All’ denotes that you must remember all the angles of the first quadrant. 

‘Subtract’ refers to subtracting one from each value of the first quadrant to get the values of the second quadrant. Each value will be placed at the corresponding value of the first quadrant. 

‘Add’ specifies adding one to the denominator of the angle of the first quadrant. The resultant will be the numerator of each corresponding angle in the third quadrant. 

‘Prime numbers’ are for the fourth quadrant. Here, each angle will begin with a prime number. 

For instance, we are comparing the angle 30and its corresponding angle 210. The value of angle 30 is 6. Adding one to the denominator to get the value of the corresponding numerator, we get the value 7. Hence, the angle 210 will be 76. 

Finding Denominator of Angle in Radians

Know the denominator of each position. Firstly, observe that denominators, other than the x and y axis, have only three numbers, i.e., 3, 4 and 6. Moreover, four is always in the middle. This means that only the positions of 3 and 6 are changing. 

So, now begin by putting six near the x-axis of the first quadrant and move upside, writing four and then three. Go forward to the second quadrant. Here, you will begin with the same denominator, which ended in the previous quadrant. Hence, the second quadrant will now start with three, followed by 4 in the middle and six at the end. Go on to the third quadrant, beginning with 6 and so on. 

Method 2: Finding Angle in Degrees

Spread your left palm so that your thumb and pinky finger are at a right angle. Now, the hand will depict the first quadrant of the circle, the sin will be the y-coordinate, and the cos will be the x-coordinate of the angle. The remaining fingers between the thumb and pinky will be represented as angles at specific degrees. Beginning from the pinky finger, it is the zero-degree angle and the x-axis. The next/ring finger will represent a 30-degree angle, followed by the middle finger at a 45-degree angle. The index finger will represent a 60-degree angle, and the thumb is a 90-degree angle or y-axis. 

To calculate sin and cos, remember that the representation of angles will be the square root of x divided by 2. The representation will be as follows:

x/2 

We have to find the value of x for each cos and sin angle. There is an easy way to remember and find it. 

Calculating cos and sin

To find the cos and sin, ratios fold the finger at a specific angle of interest. Now, you will get a few fingers at each left and right. Let’s understand this with an example of sin and cos 30. Here, you will fold the ring finger. Now, to your left, you will have three spread fingers, and to the right, you will have one spread finger. 

The left one will be your value of x for cos, and the number of fingers on the right will be your value of y for sin. So, you will get cos 30=root3/2 and sin 30=root1/2 which will be eventually sin 30=root1/2. 

The given information was for the first quadrant. Now, as you know the signs of x and y of each coordinate, simply change them while keeping the values intact. You will have a value of each angle in each quadrant of the unit circle. 

Method 3: Applying Different Tricks

You can try other methods for memorization, such as learning it through flashcards, songs or playing associated games. The games and songs are easily available with Google search. You can find flashcards online or make them yourself. We suggest making the flashcards yourself, which will help you understand and memorize the angles. 

Practice Unit Circle With Turito 

While offering methods to learn unit circles, we also suggested fun ways to practice them. Turito can unlock more of the practical applications of unit circles in depth. With emphasis on syllabus and competitive exam-based practice, the knowledge gained here will be beneficial in the long run. 

Wondering what Turito is? Turito is one of the leading educational platforms that supports and assists students in achieving their dreams and goals. Offering a one-stop self-learning solution for classes from 4 to 12 and way up for competitive exams, it provides transparency through free demo classes. Visit their website now to learn about the numerous attractive features of their website. 

Frequently Asked Questions