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.