4.3B The Unit Circle
The unit circle is a way to "standardize" the special ratios of 0o ,30o ,45o ,60o ,90o ,120o ,135o ...etc. onto one diagram/memory aid.
This is accomplished by using terminal arms in all quadrants that are 1 unit long (ie. r=1).
For 45o
(1,1)
( __ , __ )
÷___
√2
1
1
45o
45o
1
sin 45o=
cos 45o=
tan 45o=
NOTE: the coordinates
are now (cos 45, sin 45)
For 30o
(√3,1)
2
60o
1
30o
÷___
( __ , __ )
1
60o
30o
√3
sin 30o=
cos 30o=
tan 30o=
NOTE: the coordinates
are now (cos 30, sin 30)
For 60o
( __ , __ )
( 1,√3 )
÷___
30o
2
√3
1
30o
60o
60o
1
o
sin 60 =
cos 60o=
tan 60o=
Creating the Unit Circle:
start with a circle where r=1
consider the terminal arm in each quadrant
where the related angle is 45o
consider the terminal arm in each quadrant
where the related angle is 30o
consider the terminal arm in each quadrant
where the related angle is 600
NOTE: the coordinates
are now (cos 60, sin 60)
Ex. 1
Use the unit circle to determine the exact value of each
of the following.
a)
sin 120o
b)
tan 240o
c)
cos 225o
d)
cos 315o
f)
sin 300o
g)
The Unit Circle
*e) sin 310o
sinθ
cosθ
= y
x
tanθ=
Ex. 2
Use the unit circle to determine the
measure of θ, for 0 ≤ θ ≤ 360o.
sinθ =
- √3
2
b)
cos θ= 1
√2
c)
tanθ = -√3
d)
1
sinθ=
2
The Unit Circle
a)
tan 120o