Unit Circle Definition: Unit circle is a part of mathematics which is simply a circle having one radius, the concept of unit circle is basically used in the trigonometry part of mathematics where this unit circle or the circle having one radius which is centered at the coordinate ( 0, 0 ) in the Euclidean plane’s Cartesian coordinate system.
Tangent on Unit Circle
The denotation symbol for the unit circle is S¹ and Unit sphere is the generalization of the circle in higher dimensions. If we are having ( x, y ) as a point of unit circle’s circumference then in this there are two points x and y where these are the sides of a right angle triangle which is having the hypotenuse of length 1. So, now if we apply the Pythagorean theorem then we will get an equation relative to x and y will be :
Unit Circle Equation
x² + y² = 1
As the value of x² is equal to the ( – x )² for all x and all of the reflected points on a unit circle with respect to the xaxis and yaxis and the equation of the unit circle which is mentioned above can hold both of the points which are x and y on a unit circle in the first quadrant of it. The inside part of a unit circle is known as the unit disk and when the unit circle combines with itself that is with unit circle then this combination is known as the closed unit disk. There are also different types of unit circle which can be defined as by the use of distance like as of the Riemannian circle.
Unit circle in the complex plane
Unit complex number is also considered within unit circle like as of the set of the complex number z is –
z = e ͥ ͭ = cos ( t ) + i sin ( t ) = cis ( t )
This above relationship is known as the Euler’s formula. If we talk about the quantum mechanics then this equation is known as the phase factor.
Trigonometric functions on the Unit Circle
Unit Circle (sin cos tan)
There are basically two trigonometric functions which are sine and cosine for the angle θ which can be defined by the unit circle as follows : Let x and y or we can also say ( x , y ) are two different points in a unit circle and there is a ray which is passing from the origin point which is ( 0 , 0 ) up to the point ( x , y ) which is making an angle of θ from positive side of xaxis then at this time we get :
Cos ( θ ) = x
Sin ( θ ) = y
Now, there is a equation which we found above x² + y² = 1
gives a relation to these functions i.e,
cos² ( θ ) + sin² ( θ ) = 1
Now, according to the unit circle, it is also demonstrated that these trigonometric terms sine and cosine are the periodic function which are having identities :
cos θ = cos ( 2 π k + θ )
sin θ = sin ( 2 π k + θ )
where k denotes an integer.
Unit Circle Test
On the unit circle we sometime see the construction of triangles over it which are mostly used to show the periodicity of trigonometric functions for this we have to first of all make a radius named as OA coming from the origin to any point P ( x ₁ , y ₁ ) on a unit circle in such a way that it will make an angle of t with respect to 0 < t < π / 2 and this will form from the positive side of x axis. Now , let us take any other point which is Q ( x ₁ , 0 ) and draw a line segment from the PQ to OQ which is perpendicular.
After completing their we will got a right angles triangle in result of it which is Δ OPQ having angle QOP = t. Now the PQ will be length as y ₁ , OQ is having the length as x ₁ and OA will be equal to 1 , here sin ( t ) will be equal to y ₁ and tell last cos ( t ) is equal to x ₁. Now, after completing all these measurements we are going to draw another radius from the origin point to the other point on the unit circle which is R ( – x ₁ , y ₁ ) named as OR which the same angle of t which will be created from the negative side of xaxis.
Now, let us assume any other point which is S ( – x ₁ , 0 ) on the unit circle with a line segment RS perpendicular to the OS. In result of all these constructions, we will get another right angled triangle Δ ORS with the angle SOR equal to them. So , now we will see that as the angle ROQ is equal to the π – t , and R is situated at the ( cos ( π – t ) , sin ( π – t ) ) similarly we will have P at the ( cos ( t ) , sin ( t ) ).
Unit Circle with Everything
When we work with the Β right angle triangles then the periodic functions sine, cosine and other trigonometric functions will only take ten measurement which is more than zero and is less than the π / 2. If we define these functions with the unit circle then they will produce a worth full value for any real value angle measurement even with those which are greater than 2 π. All of treatment six trigonometric functions that are sine, cosine, tangent, secant, cosecant, cotangent are well archaic function like as of the versine and exsecant and these are easy to be defined geometrically with the term of unit circle. We can also calculate the value of any six trigonometric functions or for many other angles with help of this unit circle without even using any calculator for calculating values or without even using sum and difference formulas.
Circle Group of Unit Circle
With the help of Euclidean plane we can find or identify different complex numbers with help of points mentioned over it which are named as a + bi and are mostly identified by the points ( a , b ). During this identification process the unit circle comes with a group within multiplication this group is known as the circle group which is mostly denoted by the symbol T. On this plane the multiplication of terms cos θ + sin θ provides a counter clockwise rotation with respect to θ. This group is very important and has lots of applications in the field of mathematics and science.
Printable Unit Circle
Complex Dynamics of Unit Circle
A set of the discrete nonlinear dynamical system with the evolution factor is known as the unit circle as it is a very simple case so this below equation is mostly used for the study of dynamical systems :
f ̥ ( x ) = x²
Unit Circle Chart
Now , here we are going to share unit circle chart which is also a very important part of unit circle and needs to memorize it properly for finding values of some essential elements therefore here we are providing you the full chart of unit circle which can be downloaded and you can keep it with you for your remembrance.
Unit Circle Radians
Now, we are going to talk about unit circle radian. The measurement of angles in degree is from the ancient time and said that there are total 360 degrees in a circle as we are here talking about radian so it is basically a measurement of angles which are based on the different characteristics of a circle as a normal it is found that there are total 2 π of radian in a unit circle , as the unit circle trig is having one radius therefore the circumference of it is 2 π . So , this 2 π radian is equal to the 360° which means that 1 radian is equal to the 180 / π degree and 1 degree is equal to the π / 180 radian.
The reason of using it as that there are many formulae and this make it easy to write and understand when we measure angles in radian let us give an example of it , If A and B are two different points present on any circle having radius R and a center C then the arc length of this circle connecting both of them will be given by the :
d ( A , B ) = R a
here R is the radius of this circle and a is the angle of ACB which is measured in the unit radian and if on other hand we measure it in degree then we will get the formula like
d ( A , B ) = R a π / 180
You can also check the following formulas by simply having notice that the arc length of circle is proportional to the angle made by it then check it for the whole circle and at end we will get 2 π or 360 degrees.
Unit Circle with Tangent
Sin ( θ )

Cos ( θ )  Tan ( θ ) 
Sin ( 0 ) ° = 0  Cos ( 0 ) ° = 1  Tan ( 0 ) ° = 0 / 1 = 0 
Sin ( 30 ) ° = 1 / 2  Cos ( 30 ) ° = √ 3 / 2  Tan ( 30 ) ° = 1 / 2 x √ 2 / 3 = √ 3 / 3 
Sin ( 45 ) ° = √ 2 / 2  Cos ( 45 ) ° = √ 2 / 2  Tan ( 45 ) ° = √ 2 / 2 x 2 / √ 2 = 1 
Sin ( 60 ) ° = √ 3 / 2  Cos ( 60 ) ° = 1 / 2  Tan ( 60 ) ° = √ 3 / 2 x 2 / 1 = √ 3 
Sin ( 90 ) ° = 1  Cos ( 90 ) ° = 0  Tan ( 90 ) ° = 1 / 0 = undefined 
Tangent function is basically a periodic function which is a very crucial part of trigonometry in mathematics. Unit circle tan is one of the very easy ways for understanding a tangent function so here we are going to discuss unit circle with tangent. Let us draw a unit circle with the angle measurement as θ on a coordinate plane and then we will draw an angle which is centered at the origin of a circle which is having one side of positive xaxis.
Now, the Coordinate x of this point at which the other side of the angle is intersecting with the circle is known as cos ( θ ) and the y coordinate of this circle will be known as sin ( θ ). Before going further we have to remember some sine and cosine values which are important and based on the 30 degree , 60 degree , 90 degree triangles and for the 45 degree, 45 degree, 90 degree triangle as with help of we can find out the values for unit circle tangent which are related to these functions. So, the table is given above.
How to memorize Unit Circle
1 . The angles which are having their terminal arm in Quadrant II will have positive value for sine and the negative value for cosine, so the tangent will be negative.
2 . The angles which are having their terminal arm in Quadrant III will have negative value for sine and negative value for cosine so the tangent will be positive.
3 . The angles which are having their terminal arm in Quadrant IV will have the negative value for sine and a positive value for cosine, so the tangent will be negative.
Now, we can easily plot these found points on the Cartesian plane for showing a part of function or text part in between of 0 and 2 π.
You can find out the values for θ with help of reference angle as the value of θ will be less than 0 and greater than 2 π.
Blank Unit Circle
Here we are going to provide you a blank unit circle which will help you a lot as it is blank so you can easily do your calculations, draw graphs , unit circle tangents etc and can also use it for your professional work , school assignment , homework or even for your daily based practice as you are going to get it absolutely free you just have to download and take the print of this blank unit circle which will automatically become a record and solve lots of your trigonometric problems related to the unit circle.
Many of times it happens that we find difficult to draw a regular unit circle and for your easy and to save time here we are providing you this blank unit circle. There are also various sites where you can get unit circle quiz, here in this article we are also providing you one quiz for more practice of yours.
Unit Circle Table
For solving any trigonometric problem it is very necessary to know about the values of sine, cosine etc and this correct information can only be given by the table of unit circle so here we are going to provide you this complete table for your reference and easy memorization.
Degrees

Sin θ  Cos θ  Tan θ  Cot θ  Sec θ  Cosec θ 
0 °  0  1  0  undefined  1  undefined 
30 °  1 / 2  √ 3 / 2  √ 3 / 3  √ 3  2 √ 3 / 3  2 
45 °  √ 2 / 2  √ 2 / 2  1  1  √ 2  √ 2 
60 °  √ 3 / 2  1 / 2  √ 3  √ 3 / 3  2  2 √ 3 / 3 
90 °  1  0  undefined  0  undefined  1 
120 °  √ 3 / 2  – 1 / 2  – √ 3  – √ 3 / 3  – 2  2 √ 3 / 3 
135 °  √ 2 / 2  – √ 2 / 2  – 1  – 1  – √ 2  √ 2 
150 °  1 / 2  – √ 3 / 2  – √ 3 / 3  – √ 3  – 2 √ 3 / 3  2 
180 °  0  – 1  0  undefined  – 1  undefined 
210 °  – 1 / 2  – √ 3 / 2  √ 3 / 3  √ 3  – 2 √ 3 / 3  – 2 
225 °  – √ 2 / 2  – √ 2 / 2  1  1  – √ 2  – √ 2 
240 °  – √ 3 / 2  – 1 / 2  √ 3  √ 3 / 3  – 2  – 2 √ 3 / 3 
270 °  – 1  0  undefined  0  undefined  – 1 
300 °  – √ 3 / 2  1 / 2  – √ 3  – √ 3 / 3  2  – 2 √ 3 / 3 
315 °  – √ 2 / 2  √ 2 / 2  – 1  – 1  √ 2  – √ 2 
330 °  – 1 / 2  √ 3 / 2  – √ 3 / 3  – √ 3  2 √ 3 / 3  – 2 
360 °  0  1  0  undefined  1  undefined 
Unit Circle Calculator
Unit circle calculator is basically an online calculator which is being used widely for the calculation of given angles for the unit circle in the term of radian , With help of this calculator we can easily calculate the values or angles of sine, cosine and tan for any particular point. There is a relation which is given in mathematics in between of radian and angle , it is :
Rad = x ° * π / 180 ( here , x is the given angle in terms of degree )
So , here we are going to tell you that ho you can calculate the values for unit circle with help of online calculator :
Step – 1 : First of all you have to go through the online unit circle calculator which are available on web or you can also click on the below provided link
http://calculator.mathcaptain.com/unitcirclecalculator.html
Step – 2 : After clicking on the above link you will be automatically redirected to the site of online calculator, here you will a blank box which is the first box in which you have to put the value of angle of unit circle in terms of degree.
Step – 3 : Now , after substituting the value you have to click on the option of Solve which is present just below it.
Step – 4 : Wait for few seconds and you will get your answer in the below provided boxed in terms of each unit.
How to memorize the Unit circle
A good knowledge of unit circle will not only help you in trigonometry or geometry instead it will also help you in future for solving calculus , when we talk about remembering the unit circle it seems that there is a lot of data to memorize but if you understand it properly then it is very simple you have to follow just some tricks let us come to know how you can memorize a unit circle. For memorizing the radian measurement of a unit circle you have to first of all draw a circle with 2 lines which intersect each other at the center of circle now start from the East of circle and label it as 2 π then after go around this circle and label every point as π / 2 , π and 3 π / 2 respectively. Let us learn it more deeply :
Unit Circle pdf
Memorizing radian [memorizing the unit circle]
1 . First of all you have to draw two perpendicular lines which needs to intersect each other at the center of circle and these both lines will be x axis and y axis of graph.
2 . Now, with help of compass draw a large circle keeping the intersection point as origin of the circle.
3 . Now , we have to understand radian properly , as we know that a radian is a measurement of an angle. So the radian measurement of unit circle should always be assume to start from the coordinates ( 0 , 1 ).
4 . You should keep in mind that the circumference of circle is equal to the 2 π r where r is equal to the radius of circle , as the value of radius for unit circle is 1 therefore it’s circumference is equal to 2 π .
5 . Now , we have to label all four points present on the x axis and y axis , so let us divide all four parts in four directions and then we will label them –
* East : It is the starting point so you have 0 radian.
* North : As you have covered 1 / 4 part of circle, therefore, radian will be equal to 2 π / 4 = π / 2 radian.
* West : As we have covered half of the circle, therefore, value will be equal to π radian.
* South : As we completed three quarters so the radian will be equal to 2 π x 3 / 4 = 3 π / 2 radian.
6 . Next we are going to divide this circle into eight parts so draw a line from each quadrant dividing it into two equal halves and the values will be for the divided quadrants are :
* π / 4
* 3 π / 4
* 5 π / 4
* 7 π / 4
7 . Now, again we are going to divide this circle into six segments so you have to draw so lines which will cut circle in six equal segments and label these measurements for each quadrant.
* π / 3
* 2 π / 3
* 4 π / 3
* 5 π / 3
8 . Next , we are going to draw this circle in twelve parts and the last point of this unit circle will represent one twelfth part of the circumference.
* π / 6
* 5 π / 6
* 7 π / 6
* 11 π / 6
Memorizing x , y coordinates (cosine, sine )
For more information click here.
1 . First of all you are required to properly understand about sine and cosine , as we know that the basic importance of unit circle is with right angle trigonometry , every x coordinate of the circle is always equal to the value of cosine θ and every y coordinate of circle equals to the value of sine θ where the θ is value of particular angle.
2 . Next , we have to mark the coordinates for all four points of the unit circle. As a unit circle composed of only one radius and is being used for finding the value of x and y coordinates for all four points of the circle at which it gets intersect with axis. So we area gain going to take directions for remembering coordinates of it
* The coordinate of East point will be ( 1 , 0 ).
* The coordinates of North point will be ( 0 , 1 ).
* The coordinates of West point will be ( – 1 , 0 ).
* The coordinates of South point will be ( 0 , – 1 ).
3 . You have to memorize all of the coordinates for the first quadrant , the first quadrant is available at the upper right side of the circle at this point both x and y coordinates are positive below we are giving some coordinate value which you have to memorize for your future reference :
* For the π / 6 the coordinates are ( √ 3 / 2 , 1 / 2 )
* For the π / 4 the coordinates are ( √ 2 / 2 , √ 2 / 2 ).
* For the π / 3 the coordinates are ( √ 1 / 2 , √ 3 / 2 ).
4 . Next we are going to draw a straight line which will fill other coordinates so you can easy dream horizontal or vertical lines passing through both coordinates.
5 . Now , you have to memorize sign which one is positive and which one is negative there are lot of ways by which you can recognize the signs in your circle.
* We just have to remember the basic graph rules in which we have upper part as positive and below part as negative , the right side of x axis is positive and the left side of x axis is negative.
* Now , you have start drawing a line from quadrant 1 if it crosses the y axis then value of y will switch sign and if it will cross the x axis then the x value will switch sign.
* You can also memorize the rule ” All students take calculus ” ( ASTC ) and love in clockwise direction from quadrant 1 it will respectively show that :
– All values in the quadrant 1 will be positive.
– All values in the quadrant 2 will be positive only for sine.
– All values in the quadrant 3 will be positive for tangent only.
– All values in the quadrant 4 will be positive only for cosine.
Filled in Unit Circle