Unit circle is a circle of radius 1(unit) centered at origin(0, 0) in Cartesian coordinate system. A unit circle can be represented as shown in the following figure:

If (x, y) are the coordinates of a point P at circumference of a unit circle, then x and y will be the two sides of right triangle OAP. Using the Pythagorean theorem, the equation of the unit circle is given below:

Here 1 represents the radius of the circle.

## Trigonometric Functions on the Unit Circle

Trigonometric functions can also be defined on unit circle.

Let us assume an arbitrary point P at its circumference. If this line segment joining P to origin O subtends an angle of θ, then:

$sin\theta$ = $\frac{PA}{OP}$=$\frac{y}{1}$

⇒ y = sinθ

and

$\cos \theta$ =$\frac{OA}{OP}$=$\frac{x}{1}$

⇒ x = cosθ

Plugging above values in the equation of circle, we obtain the following relation:

$\cos ^{2}\theta +\sin ^{2}\theta =1$

## Unit Circle Coordinates

Unit circle is divided into four quadrants in Cartesian coordinate system. We can easily find different points at the circumference of a unit circle.

For example:

If we consider a point in first quadrant which is inclined at 30° from positive x-axis, then

x = cos 30$^o$ = $\frac{\sqrt{3}}{2}$

y = sin 30$^o$ = $\frac{1}{\sqrt{2}}$

Therefore, coordinates of this point are ($\frac{\sqrt{3}}{2}$, $\frac{1}{\sqrt{2}}$)

Following is the diagram showing various coordinates of sin $\theta$ and cos $\theta$ on a unit circle:

## Unit Circle Problems

Few problems based on unit circle are given below:

### Solved Examples

Question 1: Find a point which corresponds to the angle $\frac{\pi }{2}$ on unit circle.
Solution:

Let us assume that (x, y) be the point which corresponds to the angle $\frac{\pi }{2}$ on unit circle.

x = cos $\theta$ and y = sin $\theta$

Substituting $\theta$ = $\frac{\pi }{2}$

x = cos$\frac{\pi }{2}$ = 0

y = sin$\frac{\pi }{2}$ = 1

Hence the point corresponding to $\frac{\pi }{2}$ is (0,1).

Question 2: If a point (x, $\frac{-1}{2}$) lies at the circumference of a unit circle, find the value of x.
Solution:

Here, y = $\frac{-1}{2}$

The equation of unit circle is given by:

x2 + y2 = 1

Substituting the value of y in above equation:

x2 + ($-\frac{1}{2}$)2 = 1

x2 + $\frac{1}{4}$ = 1

x2 = 1 - $\frac{1}{4}$

x2 = $\frac{3}{4}$

x = $\pm \frac{\sqrt{3}}{2}$

Question 3: Prove that the point $(-\frac{1}{2},\frac{\sqrt{3}}{2})$ lies at the circumference of a unit circle.
Solution:

The equation of a unit circle is given by:

x2 + y2 = 1

Here, x = $-\frac{1}{2}$ and y = $\frac{\sqrt{3}}{2}$

Substituting above values in equation of unit circle, we get:

$(-\frac{1}{2})^{2}$ + $(\frac{\sqrt{3}}{2})^{2}$ = 1

$\frac{1}{4}$ + $\frac{3}{4}$ = 1

1 = 1

Since the equation of unit circle is satisfied by the above point, hence point $(-\frac{1}{2},\frac{\sqrt{3}}{2})$ must lie on the circle.