# Understanding the unit circle | StudyPug (2023)

## What is Unit Circle?

In the world of calculus, pre-calculus, and trigonometry, you will often find reference toward and problems regarding "the unit circle." But, oddly, we are rarely ever taught what it is!

In simple terms, the unit circle is a mathematical tool for making the use of angles and trigonometric functions easier. By understanding and memorizing "the unit circle" we are able to breeze through otherwise calculation-heavy problems, and make our lives a whole lot easier.

The unit circle, in it's simplest form, is actually exactly what it sounds like: A circle on the Cartesian Plane with a radius of exactly $1 unit$1unit. Like this blank unit circle below:

Next, by filling in this unit circle with commonly used angles and evaluating these angles with sine and cosine, we get something a $little$little more complicated:

Scared? Don't be. This image might seem intimidating, but when we break it down it more coherent parts, patterns start to emerge.

## Unit Circle With All 6 Trig Functions Chart:

Instead of referring to that intimidating image above, let's simplify the unit circle with $\sin \cos \tan \sec \csc$sincostanseccsc and $\cot$cot in a nice little chart:

This above unit circle table gives all the unit circle values for all 4 unit circle quadrants. As you can see, listed are the unit circle degrees and unit circle radians. You should know both, but you're most likely to be solving problems in radians. Now, the next natural question is, how can I remember the unit circle?

## How to Memorize the Unit Circle:

Memorizing the unit circle is actually much easier than you'd think, thanks to a couple little tricks:

Trick 1:

Because of the following 4 equations, we only need to memorize the unit circle values for sine and cosine.

$\tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \csc \theta = \frac{1}{\sin \theta}$tanθ=cosθsinθ,cotθ=sinθcosθ,secθ=cosθ1,cscθ=sinθ1

With these 4 equations, we don't even need to memorize the unit circle with tangent!

Trick 2:

By knowing in which quadrants x and y are positive, we only need to memorize the unit circle values for sine and cosine in the first quadrant, as the values only change in their sign. To use this trick, there are a few things we need to understand first:

i) The first important thing to note is what values sine and cosine give us on the unit circle. Because of SOHCAHTOA, we know this:

$\sin \theta$sinθ gives us the Y-coordinate and $\cos \theta$cosθ gives us the X-coordinate

ii) Now looking at each quadrant:

Quadrant 1: X is Positive, Y is Positive

Quadrant 2: X is Negative, Y is Positive

Quadrant 3: X is Negative, Y is Negative

Quadrant 4: X is Positive, Y is Negative

iii) Next, looking at where each quadrant lies:

Quadrant 1: 0 – $\frac{\pi}{2}$2π

Quadrant 2: $\frac{\pi}{2} - \pi$2ππ

Quadrant 3: $\pi$π$\frac{3\pi}{2}$23π

Quadrant 4: $\frac{3\pi}{2} - 2\pi$23π2π

iv) The value of sine and cosine will always be "the same" for the same denominator:

$\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} and \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$sin3π=23andsin34π=23

With these tricks in mind, the process of how to remember the unit circle becomes so much easier!

## How to Use the Unit Circle:

The best way to get comfortable with using the unit circle is to do some unit circle practice.

Example 1:

Find $\sin \frac{4\pi}{3}$sin34π

Since we're dealing with sine, which we will eventually have memorized, all we need to do is figure out what quadrant we're in so we know whether our answer will be positive or negative.

Since:

$\frac{3\pi}{2} > \frac{4\pi}{3} > \pi$23π>34π>π

We are therefore in the third quadrant. Thus, since sine gives us the y coordinate, and we are in the third quadrant, our answer will be negative!

Step 2: Solve

The next step is simple – using what we've memorized, we can easily solve this problem.

$\sin \frac{4\pi}{3}$sin34π = -$\frac{\sqrt{3}}{2}$23

Example 2:

Find $\tan \pi$tanπ

Since we're dealing with the unit circle with tan, we will need to use the values we've memorized from sine and cosine, and then solve. First, however, we need to figure out what quadrant we're in so we know whether our answers for sine and cosine will be positive or negative.

Since:

$\frac{3\pi}{2} > \pi > \frac{\pi}{2}$23π>π>2π

We are therefore in between the second and third quadrant on the x-axis. Since sine gives us the y coordinate, and we are on the x-axis, our answer will actually be zero! Also, since cosine gives us the x-coordinate, and we are in between the second and third quadrant (where cosine is negative for both), our answer will be negative!

Step 2: Solve

The next step is simple – using what we've memorized, we can easily solve this problem. But in this case, we need one extra step. We must use the equation for tangent discussed earlier in trick 1, assuming we haven't memorized the values for tangent on the unit circle.

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0}{-1} = 0$tanθ=cosθsinθ=10=0

Example 3:

Find $\csc \frac{\pi}{6}$csc6π

Since we're dealing with cosecant, it is important to recognize we will need to use sine values to solve using the equation for cosecant discussed earlier in trick 1. First, however, we need to figure out what quadrant we're in so we know whether our answer for sine will be positive or negative.

Since:

$\frac{\pi}{2} > \frac{\pi}{6} > 0$2π>6π>0

We are therefore in the first quadrant. Thus, since sine gives us the y coordinate, and we are in the first quadrant, our answer will be positive!

Step 2: Solve

The next step is simple – using what we've memorized, we can easily solve this problem. But in this case again, we need one extra step. We must use the equation for cosecant discussed earlier in trick 1, assuming we haven't memorized the values for cosecant on the unit circle.

$\csc \frac{\pi}{6} = \frac{1}{\sin \frac{\pi}{6}} = \frac{1}{0.5} = 2$csc6π=sin6π1=0.51=2

Now that we've done some practice, do some more on your own! In no time at all, you'll be ready for any upcoming unit circle quiz.

## FAQs

### Understanding the unit circle | StudyPug? ›

Memorizing the unit circle is actually much easier than you'd think, thanks to a couple little tricks: Trick 1: Because of the following 4 equations, we only need to memorize the unit circle values for sine and cosine. With these 4 equations, we don't even need to memorize the unit circle with tangent!

Is unit circle hard to memorize? ›

Memorizing the unit circle is actually much easier than you'd think, thanks to a couple little tricks: Trick 1: Because of the following 4 equations, we only need to memorize the unit circle values for sine and cosine. With these 4 equations, we don't even need to memorize the unit circle with tangent!

What are the important points on the unit circle? ›

Usually, the unit circle's centre point is the point where the x-axis and y-axis intersect, or at the coordinates (0,0). This is a triangulation concept that allows mathematicians to extend sine, cosine, and tangent by frequency outside the traditional right triangle.

Why is the unit circle 1? ›

Because the number 1 is called "the unit" in mathematics, a circle with a radius of length 1 is called "the unit circle". Once the hypotenuse has a fixed length of r = 1, then the values of the trig ratios will depend only on x and y, since multiplying or dividing by r = 1 won't change anything.

Is each point on the unit circle How do you know? ›

The unit circle is the circle of radius that is centered at the origin, . ( 0 , 0 ) . and this is the equation of the unit circle: a point lies on the unit circle if and only if . x 2 + y 2 = 1 .

Why do we read the unit circle counterclockwise? ›

It's because clockwise angles are taken to be negative while counterclockwise angles are positive. Thus to define the functions, we work on positive angles, hence counterclockwise.

What is a fun fact about unit circle? ›

Facts! The Unit Circle has a radius of 1. Where the center of the circle is the origin on a graph. The unit circle and trigonometry date back to the 2nd millennium BC to Egyptian mathematics and Babylonian mathematics.

How many circle theorems do you need to know for GCSE? ›

There are seven circle theorems. An important word that is used in circle theorems is subtend .

Why is the unit circle important in real life? ›

In the Real World - At A Glance

It shows up in architecture, engineering, geography, astronomy, digital imaging, and a host of other fields. It can be used to calculate distances like the heights of mountains or how far away the stars in the sky are.

What are the 4 important parts of a circle? ›

The 4 main parts of a circle are radius, diameter, center, and circumference. The center of the circle is the point that is equidistant from all the sides of the circle. The radius is the length of the line from the center of the circle to any point on the curve of the circle.

### What are the 4 parts of the unit circle? ›

The four quadrants on the unit circle are created by dividing a 360 degree circle into 4 equal 90 degree parts. The quadrants go from 0-90, 90-180, 180-270 and 270-360.

What are the six functions of the unit circle? ›

These six functions and their abbreviations are sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (csc), and cotangent (cot).

How do you describe a unit circle in terms of complex numbers? ›

The unit circle is the circle of radius 1 centered at 0. It include all complex numbers of absolute value 1, so it has the equation |z| = 1. A complex number z = x + yi will lie on the unit circle when x2 + y2 = 1.

Why is the unit circle 2 pi? ›

Why is a circle 2 pi radians? radians are related to the arc of a circle. one radian equals an arc length that corresponds to 1 radius in distance, But the circle has a total arc length C=2 x Pi x r. Hence a full circle in radians is 2Pir/r=2 Pi.

Top Articles
Latest Posts
Article information

Author: Errol Quitzon

Last Updated: 11/02/2023

Views: 5674

Rating: 4.9 / 5 (79 voted)

Author information

Name: Errol Quitzon

Birthday: 1993-04-02

Address: 70604 Haley Lane, Port Weldonside, TN 99233-0942

Phone: +9665282866296

Job: Product Retail Agent

Hobby: Computer programming, Horseback riding, Hooping, Dance, Ice skating, Backpacking, Rafting

Introduction: My name is Errol Quitzon, I am a fair, cute, fancy, clean, attractive, sparkling, kind person who loves writing and wants to share my knowledge and understanding with you.