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Unit Circle

As the name hints, a unit circle is a circle with a radius of one unit. A circle is a closed geometric shape that has no angles and no sides. The unit circle has all the features of a circle, and its equation is derived from a circle's equation. A unit circle is used to calculate the standard values of angles of all kinds of trigonometric ratios

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Define Unit Circle?

As the name indicates, a unit means one, and a circle is itself a circle. It is a kind of circle which is having a radius of only one unit. A unit circle is a closed geometric shape having no angles and no sides. The features of a unit circle resemble the simple circle, and its equation is also derived from a circle's equation.

A unit circle is also used to calculate the standard values of angles of all kinds of trigonometric ratios.

With the help of trigonometric ratios like cosines and sines, we will be able to solve unit circle equations. 

The second-degree equation with two variables, x, and y, represents the unit circle in algebra. The unit circle is functional in trigonometry for calculating the values of trigonometric ratios; such as sine, cosine, and tangent.

How to use this tool?

Just select the unit in which you have to do the calculations:

Ø  degrees (deg)

Ø  radians (rad)

Ø  π radians (* π rad)

Then enter your desired values, then the tool will display the required results.

What do you mean by a Degree?

The degree is the fundamental unit of angular measurement. Despite being the most often involved unit in practice, it is not the System of International (SI) unit of angular measurements. A degree, i.e., arc degree, is equal to 1/360th of a circle's entire angle.

The formula for degrees is:

Degrees = Radians × 180 / π

What is a Radian?

The plane angle subtended by a circular arc of length equal to its radius is considered as one radian.

The standard unit of angular measurement is radian, and it's employed almost everywhere in the field of mathematics. Radian has no dimension and is the derived SI unit of angular measurements. 

The formula for radians is:

Radian = Degrees * π / 180°

In the same context for

π Radian = Degrees / 180

Can I do this calculation by myself?

Yes! You can easily calculate the values of a unit circle by yourself. These are some examples explained below.

Example 1:

Let’s suppose you know the value of degrees which is 9.

Now, substituting the values in the formula above for radians,

  • For Radian

Radian = Degrees * π / 180°

R = 9 * π / 180 = 9 * 3.14 / 180 = 28.26 / 180

R = 0.157

  • For π Radian

π Radian = Degrees / 180

π R = 9 /180 = 0.05

Example 2:

Let’s suppose you know the value of radian which is 5.

Now, substitute the given values in the above mentioned formula,

  • For Degrees

Degrees = Radians × 180 / π

D = 5 * 180 / π = 900 / 3.14

D = 287

  • For π Radian

Now that we know the value of a degree from radian we can easily extract the value of π R by putting the formula:

π Radian = Degrees / 180

π R = 287 / 180 = 1.59

Note*

Keep this table in your mind:

Angle Sin Cos Tan=Sin/Cos
30° 1/2 √3/2 1/√3 = √3/3
45° 2/2 √2/2 1
60° √3/2 1/2 √3

You can easily memorise them:

For sin:

It goes like “1,2,3”

  • sin(30°)  =  √1/2  =  1/2  (because √1 = 1)
  • sin(45°)  =  √2/2
  • sin(60°)  =  √3/2

For cos:

It goes like   "3,2,1"

  • cos(30°)  =  √3/2
  • cos(45°)  =  √2/2
  • cos(60°)  =  √1/2  =  1/2

For tan:

Well, tan = sin/cos, so we can calculate it like this:

  • tan(30°) =sin(30°)/cos(30°) = 1/2 ÷ √3/2 = 1/√3
  • tan(45°) =sin(45°)/cos(45°) = √2/2 ÷ √2/2 = 1
  • tan(60°) =sin(60°)/cos(60°) = √3/2 ÷ 1/2 = √3

These values are provided herewith just for your knowledge, and you can easily find out the values of sin, cos, or tan with the help of Tools City's Unit Circle tool.