Unit Circle: Definition, Formula, Diagram and Solved Examples

Unit Circle is a Circle whose radius is 1. The center of unit circle is at origin(0,0) on the axis. The circumference of Unit Circle is 2πぱい units, whereas area of Unit Circle is πぱい units². It carries all the properties of Circle. Unit Circle has the equation x² + y² = 1. This Unit Circle helps in defining various Trigonometric concepts.

The Unit Circle is often denoted as S¹ generalization to higher dimensions is the unit sphere. Let's understand more about Unit Circle, Formula and Solved examples in detail below.

What is Unit Circle?

Unit Circle is a circle that has a radius of One(1) unit. We use the cartesian plane to draw a unit circle and a unit circle is a 2-degree polynomial with two variables. The unit circle has various applications in trigonometry and algebra and is majorly used to find the values of different trigonometric ratios such as sin x, cos x, tan x, and others.

Unit Circle Definition

In Mathematics, we define a unit circle as the locus of a fixed point that is at a distance of one unit from the center of the circle. A unit circle has a radius of one unit and hence the name unit circle.

Equation of Unit Circle

We know that the equation of any circle with center (h, k) and radius 'r' is,

(x - h)² + (y - k)² = r²

For a unit circle we know that r is 1 unit and so the equation of the unit circle is,

(x - h)² + (y - k)² = 1

Formula of Unit Circle

If the center of the unit circle is origin, i.e. (h, k) = (0, 0) then the equation of unit circle is,

x² + y² = 1

A unit circle is represented in the image added below, with center coordinate h, k and when the circle is at origin the value of h and k is zero and the radius AP is equal to 1 unit.

Trigonometric Functions Using Unit Circle

The application of the Pythagoras theorem in a unit circle can be better used to understand trigonometric functions. For this, we consider a right triangle to be placed inside a unit circle in the Cartesian coordinate plane. If we notice, the radius of this circle denotes the hypotenuse of the right-angled triangle.

The radius of the circle forms a vector. This leads to the formation of an angle, say θしーた with the positive x-axis. Let us suppose x to be the base length and y to be the altitude length of the right triangle respectively. Also, the coordinates of the radius vector endpoints are (x, y) respectively.

The right-angle triangle holds the sides 1, x, and y respectively. The trigonometric ratio can be computed now, as follows:

sin θしーた = Altitude/Hypotenuse = y/1

cos θしーた = Base/Hypotenuse = x/1

Now,

• sin θしーた = y
• cos θしーた = x
• tan θしーた = sin θしーた /cos θしーた = y/x

On substituting the values of θしーた, we can obtain principal values of all the trigonometric functions. Simillarly values of trigonometric functions at different values is found.

Unit Circle with Sin Cos and Tan

Any point on the unit circle with the coordinates (x, y), is represented using trigonometric identies as, (cosθしーた, sinθしーた). The coordinates of the radius corners represent the cosine and the sine of the θしーた values for a particular value of θしーた and the radius line. We have cos θしーた = x, and sin θしーた = y. There are four parts of a circle each lying in one quadrant, making an angle of 90°,180°, 270°, and 360°. The radius values lie between -1 to 1 respectively. Also, the sin θしーた and cos θしーた values lie between 1 and -1 respectively.

Unit Circle and Trigonometric Identities

The unit circle trigonometric identities for cotangent, secant, and cosecant can be computed using the identities for sin, cos, and tan. Conclusively, we obtain a right-angled triangle with the sides 1, x, and y respectively. Computing the unit circle identities can be expressed as, 

• sin θしーた = y/1 
• cos θしーた = x/1 
• tan θしーた = y/x
• sec θしーた = 1/x
• cosec θしーた = 1/y
• cot θしーた = x/y

Unit Circle Chart

The unit circle chart is a chart that contains the value of the trigonometric function sine and cosine for various angles. The unit circle chart for the same is added below,

Unit Circle Table

The trigonometric ratios used in the unit circle table are used to list the coordinates of the points on the unit circle that correspond to common angles.

Unit Circle Pythagorean Identities

There are three Pythagorean Identities and all of them are easily proved using the concept of the unit circle the three Pythagorean identities are,

• sin²θしーた + cos²θしーた = 1
• 1 + tan²θしーた = sec²θしーた
• 1 + cot²θしーた = cosec²θしーた

Unit Circle Complex Plane

Complex Numbers and Complex Plane are easily explained using the concept of unit circle. The equation of unit circle in complex form is,

|z| = 1

OR

x²+ y² = 1

In Euler's Form complex number is represented as,

z = e^it = cos t + i(sin t)

Read More

• Trigonometric Table
• Trigonometric Identities
• Inverse Trigonometric Identities

Solved Examples on Unit Circle

Q1: Prove that point Q lies on a unit circle, Q = [1/√(6), √4/√6]

Solution:

Given,

• Q = [1/√(6), √4/√6]

x = 1/√(6), y = √4/√6

Equation of Unit Circle is,

x² + y² = 1

LHS = (1/√(6))² + (√4/√6)²

LHS = 1/6 + 4/6 = 5/6 ≠ 1

LHS ≠ RHS

Thus, point Q[1/√(6), √4/√6] does not lie on the unit circle.

Q2: Compute tan 30^o using the sin and cos values of the unit circle.

Solution:

tan 30° using sin and cos values,

tan 30° = (sin 30°)/ (cos 30°)

• sin 30° = 1/2
• cos 30° = √(3)/2

tan 30° = 1/2/√(3)/2

tan 30° = 1/√(3)

Q3: Validate if the point P [1/2, √(3)/2] lies on the unit circle.

Solution:

Given,

P = [1/2, √(3)/2]

• x = 1/2
• y = √(3)/2

Equation of Unit Circle is,

• x² + y² = 1

LHS

= (1/2)² + (√(3)/2)²

= 1/4 + 3/4

= (1 + 3)/4 = 4/4

= 1

= RHS

Practice Questions on Unit Circle

Q1. Check If the points A (1/2, 3/2) lies on a unit circle.

Q2. Check If the points A (2, 1/2) lies on a unit circle.

Q3. Find the value of cos 240°

Q4. Find the value of tan 320°

Q5. Find the value of sin 160°