Unit Circle

Unit Circle:
What is the unit circle?
Click here to explore an interactive unit circle.
A unit circle is a circle with the radius of 1 which helps determine the sin, cos, tan of given radians. The unit circle is an easy way to look at angles using radians.

The unit circle we can help us determine sin, cos, and tan of each given radian or degree.
Lets take a simple example of π/4 (45 degrees)
On the unit circle 45 degrees is written as π/4.
Because the angle is 45 we can conclude this forms a special right triangle!

The triangle on the right is what is formed from this radian
and gives us the ability to find sin, cos, and tan.

Remembering our SOHCAHTOA, we can use this triangle to determine the sin, cos, and tan.
Find sine:

Since sin= opposite of θ/hypotenuse
sin= 1/√ 2 = √ 2 / 2  (this is the height)
Find cosine:
Since cos=adjacent of θ/hypotenuse
cos=1/√ 2 = √ 2 / 2 (this is the base length)
Find tangent:
Since tan=opposite of θ/adjacent of θ
tan=1/1= 1 

An easy way to find the cos and sin is using the Pythageorean Identity
(sin)^2+ (cos)^2= 1
so...
If sin= 1/2, what is cos?
(1/2)^2+cos^2=1
1-1/4=cos^2
3/4=cos^2 (SQUARE both sides)
to get cos=  √3/2

<-SHOWS THE SIN,COS

Problems:

1) Find sinπ/3 on the unit circle, what does it mean?

2) If cosine= √ 2/2, what is sin?

3) What is the tangent of 120 degrees?

4) Show the meaning of 3 radian on the unit circle.

5) What is the sin of π/2? 

6)Why is π/4 equal to 45 degrees?

Click for answers :)