Unit Circle

Last Updated : 11 May, 2026

A unit circle is a circle with radius 1 unit, centered usually at the origin (0, 0) on the coordinate plane.

  • Equation is x² + y² = 1; every point (x, y) on it is exactly 1 unit away from the center.
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The unit circle is a key tool in trigonometry because it connects angles with coordinates:

For any angle θ (measured from the positive x-axis):

  • x = cos⁡(θ)
  • y = sin⁡(θ)

So each point on the circle represents the cosine and sine of an angle.

Unit Circle with Sin, Cos, and Tan

A unit circle can be used to understand trigonometric functions. For this, we consider a right triangle to be placed inside a unit circle. Since the radius of a unit circle is 1, it becomes the hypotenuse of the triangle.


Now,

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

On substituting the values of θ, we can obtain principal values of all the trigonometric functions.

Unit Circle Chart

The unit circle chart is a chart that contains the values of the trigonometric functions sine and cosine for various angles.

Unit-Circle

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.

Angles

0°

30°

45°

60°

90°

sin

0

1/2

1/√(2)

√3/2

1

cos

1

√3/2

1/√(2)

1/2

0

tan

0

1/√(3)

1

√(3)

Not Defined

csc

Not Defined

2

√(2)

2/√(3)

1

sec

1

2/√(3)

√(2)

2

Not Defined

cot

Not Defined

√(3)

1

1/√(3)

0

Solved Examples

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° 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

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

Q2. Check If the point 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°.

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