unit circle is a circle whose radius = 1 unit and center is (0,0). Angles measured in clockwise will have positive values and angles measured in anticlockwise will have negative values.

 

 

On unit circle, trigonometric functions are given as,

X =   Cos`theta`     and  Y =   sin `theta` 

The value of sine is on the y coordinate of the unit circle and value of cosine is on the x - coordinate of the unit circle. Using the unit circle, any trigonometric function value can be calculated by using certain formulas. We can divide the unit circle into 4 quadrants.

 

Unit Circle Diagram

 

                                                            Unit circle

We can easily make note of some angles by looking at the unit circle.

Y = sin`theta`   and X = cos `theta`

Therefore,       cos 0o = 1               sin0o= 0

                     cos 30o = `sqrt(3)` /2      sin30o  = 1/2

                     cos 45o = `sqrt(2)` /2      sin45o = `sqrt(2)` /2

                     cos 90o   =  0         sin 90 = 1

With these angles, we can also calculate other angles like tan`theta` , csc`theta`  , sec`theta` and cot`theta`

For example, to find the value of  tan 215o, we know tan`theta`  =  sin`theta` / cos `theta`

                               tan 225o  =  sin 225o/ cos 225o
                                              =   -`sqrt(2)` /2 `//` - `sqrt(2)` /2
                                              = 1
Similar way, many unknown angles can be found.

 

Properties of Unit Circle

 

  • In the first quadrant, from 0o- 90o both cos and sin are positive. Hence, all trigonometric functions are positive.
  • In the second quadrant, only sin is positive. Hence, all trigonometric functions except sin and its inverse csc are negative.
  • In the third quadrant, only tan is positive. Hence, all trigonometric functions except tan and its inverse cot are negative.
  • In the fourth quadrant, cos and sec  is positive rest all are negative.

 

Examples:

 

Example 1:

Find the area of an unit circle?

Solution:

We know that for an unit circle

radius =1 and center is (0,0)

Area = `pi r^2`

Here  r=1 and `pi` = 3.14

So the area of an unit circle is

 `= pi r^2`

 `= pi *1^2`

 `= pi * 1`

 `= pi`

 `= 3.14`

 

Example 2:

Find the equation of an unit circle?

Solution:

We know that for an unit circle

radius =1 and center is (0,0)

the genaral equation of a circle isgiven by

`x^2 + y^2 = r^2`

here r =1

`x^2 + y^2 = 1`

This is the equation of an unit circle.