The Unit CircleBy: Sarah Allen

EDU 653 The Read/Write Web

Unit Circle

A unit circle is a circle that has a center at the origin with a radius of one. The equation of a unit circle is x 2 + y 2 = 1.

Using radian measure, we can label the points on the unit circle that correspond with the degree measure.

Radian

The radian  is a unit of plane angle, equal to 180/ π degrees, or about 57.2958 degrees. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level.

http://www.reference.com/browse/Radian

Degree to Radian

Multiply the degree by π/180 and simplify but leave it in π form.

Example

6

5

18

15

180

150

180150

Radian to Degree

Multiply the radian by 180/π and simplify until you are left with only a degree measure.

Example

3003

900180

3

5

Basic Example of Unit Circle

0º = 0 90º = π/2 180º = π 270º = 3π/2 360º = 2π

Picture made by Sarah Allen

Special Right Triangles

Picture made by Sarah Allen

Building The Unit Circle

Using the special right triangles, we can start to build the points on the circle.

Picture made by Sarah Allen

Building The Unit Circle Cont.

We now know that a 30º angle has a x value of square root of three divided by two and a y value at one half. We can now convert degrees to radian and complete that point on the circle.

618

3

180

30

18030

Building The Unit Circle Cont.

Based on our results, we found that at π/6 we have the point square root of three divided by two and one half. We can continue this around entire unit circle, which is explained in the video on the next slide.Picture made by Sarah Allen

Video Building The Unit Circle

Melodyeducate (2011, Feb 8) Building The Unit Circle. Retrieved October 29, 2012 from http://www.youtube.com/watch?v=BXLxl6YRvdc

Using The Unit Circle

Now that we know how to built the unit circle we can begin to understand what it was built for.

cosine. and sineboth find to thisusecan and

2

1,

2

3point thehas

6 that know We

Sine and Cosine

Cos = adj/hyp and in the unit circle that is x/1 which is just x

Sin = opp/hyp and in the unit circle that is y/1 which is just y

Thereforecos = x and sin = y and any point on the

unit circle can be seen as ( cos, sin )

Sine and Cosine Cont.

2

1

6sin and

2

3

6cos

Therefore…

Unit Circle Game

There is a game for the unit circle that can be used for helping in the memorization of the unit circle information. Go to the following link to use this game to help you remember the information you just learned. Click on the unit circle in the lower right hand corner to follow the link.*must have an internet connection to visit the game

Picture made by Sarah Allen

Felliax08. (2007). Unit circle. Retrieved October 29, 2012 from http://www.purposegames.com/game/unit-circle-quiz/info

Practice

1. Convert 50º into radian.2. Convert 5π/7 into degree.3. What is the sin (π/3)?4. What is the cos (5π/6)?5. What is the coordinate point of 5π/4

Write your answers onto a sheet of paper and turn in to Mrs. Allen by tomorrow.