Unit Circle (original) (raw)
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.
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 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°.