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:

x^{2} + y^{2} = 1

Substituting the value of y in above equation:

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

x^{2} + $\frac{1}{4}$ = 1

x^{2} = 1 - $\frac{1}{4}$

x^{2} = $\frac{3}{4}$

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:

x^{2} + y^{2} = 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.