Table of Contents

Last modified on April 25th, 2024

# Unit Circle

## What is a Unit Circle

A unit circle is a circle of unit radius, which means it has a radius of 1 unit. It is generally represented in the Cartesian coordinate system. The unit circle has its center at the origin (0, 0).

The primary purpose of the unit circle is that it makes other functions of mathematics easier. For example, in trigonometry, the unit circle at any angle uses the value of sine and cosine.

## Unit Circle Equation

The unit circle has all the properties of a circle and its equation is also derived from the equation of a circle.

The general equation of a circle in standard form (x – h)2 + (y – k)2 = r2, which represents a circle having the center (h, k) and the radius r.

The above equation can be simplified to represent the equation of a unit circle. Since, a unit circle has its center at (0, 0) and a radius of 1 unit, substituting the values in the above equation, we get

(x – 0)2 + (y – 0)2 = 12

x2 + y2 = 1

The above equation satisfies all the points lying on the circle in all four quadrants.

## Finding the Angles of Trigonometric Functions Using a Unit Circle: Sin, Cos, Tan

We can calculate the trigonometric functions of sine, cosine, and tangent using a unit circle. Here we will use the Pythagorean Theorem in a unit circle to understand the trigonometric functions.

Let us take a point P on the circumference of the unit circle whose coordinates be (x, y).  Being a unit circle, its radius ‘r’ is equal to 1 unit, which is the distance between point P and center of the circle. By drawing the radius and a perpendicular line from the point P to the x-axis we will get a right triangle placed in a unit circle in the Cartesian-coordinate plane. The radius of the unit circle is the hypotenuse of the right triangle, which makes an angle θ with the positive x-axis. The lengths of the two legs (base and altitude) have values x and y respectively.

Thus we have a right triangle with sides measuring 1, x, y. Applying this values in trigonometry, we get

sin θ = Altitude/Hypotenuse = y/1

cos θ = Base/ Hypotenuse = x/1

tan θ = Altitude/ Base = y/x

cosec θ = 1/sin θ = Hypotenuse/ Altitude = 1/y

sec θ = 1/cos θ = Hypotenuse/ Base = 1/x

cot θ = 1/tan θ = Base/ Altitude = x/y

Since, the equation of a unit circle is given by x2 + y2 = 1, where x = cos θ and y = sin θ, we get an important relation:

sin2 θ + cos2 θ = 1

### Sign of Trigonometric Functions

The sign of a trigonometric function depends on the quadrant that the angle is found. To help remember which of the trigonometric functions are positive in each quadrant, we can use the mnemonic phrase ‘All Students Take Calculus’ or All Sin Tan, Cos (ASTC).

Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating in counterclock-wise manner. In quadrant I, which is ‘A’ all of the trigonometric functions are positive. In quadrant II, ‘Smart’, only sine is positive. In quadrant III, ‘Trig’ only tangent is positive. Finally, in quadrant IV, ‘Class’ only cosine is positive.

If we take a close look at the unit circle, we will find that the sin and cos values of angles fluctuate between -1 and 1. They take on the value of 0 as well as positive and negative values of three numbers √3/2, √2/2, and ½. Identifying the reference angles will help us identify a pattern in these values. Refer our

### Value of Angles in Trigonometric Functions

The value of θ can be expressed in both degree and in radians. It is important to remember the values of sin, cos, and tan for some specific angles. These angles are called reference angles.

In the unit circle, we have cosine as the x-coordinate and sin as the y-coordinate. Let us find their values for θ = 0° and θ = 90°.

For θ = 0°, the x-coordinate is 1 and the y-coordinate is 0. Thus, cos 0° = 1 and sin 0° = 0. Again, for θ = 90°, cos 90° = 1 and sin 90° = 1. Similarly, we can find the values of the trigonometric functions for some specific values of θ such as 30°, 60°, and 90°. These angles, called special angles, are used for simplifying the calculations related to trigonometric functions with different angles. Their value is always between 0 and 90° when measured in degrees or 0 and π/2 when measured in radians. A reference angle always uses an x-axis as its frame of reference.

The values of the special angles of all the trigonometric functions are represented in the form of a table given below.

## Preparing the Complete Unit Circle Chart

The entire unit circle represents a complete angle of 360°, which is equal to 2π when expressed in radians.

Thus mathematically,

2π radians = 360°

This means that, 1 radian = 360°/2π = 180°/π

This relation is used to convert angles from radians to degrees.

A unit circle is divided into four quadrants making an angle of 90°, 180°, 270°, and 360° (in degrees) or  π/2, π. 3π/2, and 2π (in radians) respectively.

An angle on a unit circle is always measured from the positive x-axis, with its vertex at the origin. Its initial side is on the x-axis, while the ray that begins at the origin and coincides with the point on the unit circle forms the terminal side. An angle has a positive value if it is measured by going in anticlockwise direction from the x-axis.

Thus the value of sin (x) and cosine (y) of any angle is based on the coordinate it belongs. Given below is the list:

Quadrant 1 – (0 – 90°(π/2)) : X is positive, Y is positive

Quadrant 2 – (90°(π/2) – 180°(π)) : X is negative, Y is positive

Quadrant 3 – (180°(π) – 270°(3π/2)) : X is negative, Y is negative

Quadrant 4 – (270°(3π/2) – 360°(2π) ) : X is negative, Y is negative

Next, we present some commonly encountered angles along with the special angles on the unit circle in the form of a chart combined together in degrees and in radians.

The above chart of the unit circle can also be separately represented in degrees and in radians for the sake of simplicity. They are given below.

## How to Use the Unit Circle

The best way to use the unit circle is to find the unknown angle of the trigonometric functions. Let us solve some problems to clear your concept even better.

Find the value of sin 4π/3.

Solution:

Since the given trigonometric function is sine, we just need to find in what quadrant it is in to understand whether the answer will be positive or negative.
3π/2 ˃ 4π/3 ˃ π
Thus, 4π/3 lies in the third quadrant and since sine denotes the y-coordinate, 4π/3 is negative.
Now, finding the value of sin 4π/3 = -√3/2

Find the value of sin (150°) using the unit circle.

Solution:

Since sin value is positive in the second quadrant and is denoted by y-coordinate, so we will take the second coordinate in the unit circle.
Thus, sin (150°) = ½

Find the exact value of tan 210° using the unit circle.

Solution:

As we know,
tan 210° = sin 210°/cos 210°
Using the unit circle chart, we get
sin 210° = -1/2
cos 210° = -√3/2
Thus,
=> tan 210° = -1/2/-√3/2
=> tan 210° = 1/√3

## Graphing Sine, Cosine, and Tangent Functions

Sine and cosine functions relate real number values to the x and y-coordinates of a point on the unit circle. Now, let’s learn what do they look like on a graph on a coordinate plane?

For graphing the sine functions, let’s start with the sine function: y =sin x

We can create a table and then plot the values in the graph. Given are some of the values for the sine function on a unit circle, with the x-coordinate being the angle in radians and the y-coordinate being sin x.

(0, 0), (π/6, ½), (π/4, √2/2), (π/3, √3/2), (π/2, 1), (2π/3, √3/2), (3π/4, √2/2), (5π/6, 1/2), (π, 0)

Plotting the points from the table and continuing along the x-axis to get the shape of the sine function.

Note that sine values are positive between 0 and π, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2π, which correspond to the values of the sine function in quadrants III and IV on the unit circle.

Similarly, for graphing the cosine functions, let us look at the cosine function:  y = cos x

We will follow the same procedures as sine function: create a table of values and use them to sketch a graph. Given below are the values for the cosine function on a unit circle between 0° and 180°, with the x-coordinate being the angle in radians and the y-coordinate being cos x:

(0, 1), (π/6, √3/2), (π/4, √2/2), (π/3, 1/2), (π/2, 0), (2π/3, -1/2), (3π/4, -√2/2), (5π/6, -√3/2), (π, -1)

Similar to the sine function we can plot the given points to get a graph of the cosine function.

For graphing the tangent functions for the special angles, we cannot use the unit circle. Instead we need to apply the formula tan x = sin x/cos x to determine the tangent of each value. It can be graphed by plotting (x, f(x)) points.

Let’s look at the graphical behavior of the tangent function and their values for some special angles. Given below are the values for the tangent function for which the x-coordinates are angles in radians, and the y-coordinates are tan x:

(-π/2, undefined), (-π/3, -√3), (-π/4, -1), (-π/6, -√3/3), (0, 0), (π/6, √3/3), (π/4, 1), (π/3, -√3), (π/2, undefined)

The above points will help us draw our graph, but we need to determine how the graph behaves where it is undefined.

Consider the last four points. We can see that that the value of y increases when x increases as it approaches π/2. Similarly the value of y decreases as it approaches -π/2. Now, there are multiple values of x that gives cos x = 0, which are exactly the points where tan x is undefined. At points where tan x is undefined there will be discontinuities in the graph. At such values (x = π/2, and x = – π/2) the tangent functions has vertical asymptotes.

Last modified on April 25th, 2024