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Unit Circle

In geometry, we study about circle. The circle is a vast topic which is utilized everywhere not only in mathematics, but in other fields too. The circle is defined as a locus of a point which moves at equal distance from a fixed point. In other words, it is set of points that are equidistant from a particular fixed point. This fixed point is termed as the center of the circle and the fixed distance is called as the radius of the circle.

In mathematics, the concept of a unit circle plays a vital role. A unit circle is a circle whose radius = 1 unit and center is (0,0). In other words, a circle which has unit radius and whose center lies at the origin, is known as a unit circle. Angles measured in clockwise will have positive values and angles measured in anticlockwise will have negative values.

The equation of a unit circle is :
$x^{2}+y^{2}$ = 1

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 the 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. 

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Unit Circle Diagram

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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         $ \frac{\sqrt 3}{2}$
 sin30o          $\frac{1}{2}$
  cos 45o          $\frac{1}{\sqrt 2}$
  sin45o         $\frac{1}{\sqrt 2}$
 cos 90o        0
sin 90o
      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`$\frac{sin \theta}{cos \theta}$

tan 225o$\frac{sin 225^{\circ}}{cos 225^{\circ}}$

               = 
$\frac{\frac{\sqrt 2}{2}}{\frac{\sqrt 2}{2}}$

               = 
$1$
Similar way, many unknown angles can be found. 

Properties of Unit Circle

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  • 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.

Unit Circle Examples

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Below are some examples on unit circle

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.

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