Download Table 1

Date: 1.14.10 sine function meets unit circle
At the end of the day students will…use unit circle to determine the output of the sine function for increments
of π/6…without a calculator.
Anticipation of next steps…
.
Step by Step Instruction…
1) Quiz: find the coordinates for the circle
5
 0, 1 


3 1
, 
 
2
2

6

,
1
 -1 ,0 
 1, 0 
 0, -1 
2) Find coordinates of this one:
 0, 1 

1
,


3 1
, 
 
2
2


6
 -1 ,0 
 1, 0 
 0, -1 
3) Show how the 30-60-90 triangle sides can also be found using Sine. Sinθ=opp/hyp
Make connection of sin30 = 0.5
and sin(π/6) = 0.5
FST
Unit Circle Investigation
Name:_________________________ Per:___
KEY
The circle below is called the “Unit Circle” because it has a radius of one unit.
In this investigation we will use the things we have been learning about special right triangles to help us find the
coordinates of special points on the unit circle.
Part 1. The figure below shows the points where the unit circle intersects the x and y axes. To measure an
angle  on the unit circle, we measure from the initial side of the angle, which starts at the origin and extends
along the positive x-axis (due east), counterclockwise to the other (terminal) side of the angle. So if we wanted
to measure how big the angle whose terminal side passes through point D is, we would start measuring from
segment OA counterclockwise three-fourths of the way around the circle until we reached segment OD. That
would be 270! Use this process to fill out the missing angle measures in Table 1. Then find the coordinates of
points A-D and finish filling in Table 1, using your calculator to find the values of sine, cosine, and tangent.
B
1
0.8
0.6
0.4
0.2
C
-1.5
A
-1
-0.5
0.5
1
1.5
-0.2
O
-0.4
-0.6
-0.8
-1
D
Table 1
Fill in the table below:
Point
A
B
Angle 
(Degrees)
0º
90º
Angle 
(Radians)
0
C
D
180°
270°
π/2
3π/2

2
X
Y
1
0
0
1
-1
0
0
-1
Cos  Sin 
Tan 
(exact) (exact) (exact)
1
0
0
undefined
0
1
-1
0
0
-1
0
undefined
Part 2. The next figure shows 12 more important points on the unit circle, each 30 away from either the x- or y-axis. Using the
process described in part 1, find the measure of the angles that pass through the points E-M on our unit circle and fill in their values in
Table 2.
1
G
F
0.8
0.6
H
E
0.4
0.2
-1.5
-1
-0.5
0.5
1
1.5
-0.2
O
-0.4
M
J
-0.6
-0.8
K
L
-1




Next, draw a right triangle with hypotenuse OE and with one leg along the x-axis. What special type of right triangle is
formed?
Use your knowledge of this special right triangle to find the exact values of the x and y coordinates of point E on the unit
circle.
Now, use your right triangle trig knowledge to find the exact values for the sine, cosine, and tangent of your angle.
Repeat this process for points F-M until you are able to complete Table 2. Be careful with your positive and negative signs!
Table 2
Fill in the table below:
Point Angle 
Angle
(Degrees)
X

Y
Cos

Sin

Tan

(Radians) (exact) (exact) (exact) (exact) (exact)
E
30º
π/6
F
60°
π/3
G
120°
2π/3
H
150°
5π/6
J
210º
7π/6
K
240°
4π/3
L
300°
10π/6
M
330°
11π/6
3
2
1
2
1

2
3
2
3

2
1

2
1
2

3
2
1
2
3
2
3
2
1
2
1

2
3
2
3

2
1

2

3
2
1
2
1

2
3
2
3

2
1

2
1
2

3
2
1
2
3
3
3
2
3
2
1
2
1

2
3
3
2
3

2
1

2
3

3
3
3
3
3

 3

3
3
Part 3. The last unit circle shows 4 more important points on the unit circle, each 45 away from either the x- or
y-axis. Using the process described in part 1, find the measure of the angles that pass through the points N-R on
our unit circle and fill in their values in Table 3.
1
0.8
P
N
0.6
0.4
0.2
-1.5
-1
-0.5
0.5
1
1.5
-0.2
O
-0.4
-0.6
Q
R
-0.8
-1




Next, draw a right triangle with hypotenuse ON and with one leg along the x-axis. What special type of
right triangle is formed?
Use your knowledge of this special right triangle to find the exact values of the x and y coordinates of
the Point N on the unit circle.
Now, use your right triangle trig knowledge to find the exact values for the sine, cosine, and tangent of
your angle.
Repeat this process for points P-R until you are able to complete Table 3. Be careful with your positive
and negative signs!
Table 3
Fill in the table below:
Point Angle  Angle

X
Y
Cos

Sin

Tan

(Degrees) (Radians) (exact) (exact) (exact) (exact) (exact)
N
45º
π/4
P
135°
3π/4
Q
225°
5π/4
R
315°
7π/4
2
2
2

2
2

2
2
2
2
2
2
2
2

2
2

2
2
2
2

2
2

2
2
2
2
2
2
2
2

2
2

2
1
-1
1
-1
Part 4.
Fill in the table below, assuming the circle in Part 3 now has a radius of 2 units,
instead of a radius of 1 unit.
Point Angle  Angle 
X
Y
Cos  Sin  Tan 
(Degrees) (Radians) (exact) (exact) (exact) (exact) (exact)
N
45º
π/4
P
135°
3π/4
 2
Q
225°
5π/4
 2
R
315°
7π/4
2
2
2
2
 2
 2
2
2
2

2
2

2
2
2
2
2
2
2
2

2
2

2
1
-1
1
-1
What values changed? Why?
X, Y, because the hypotenuse is 2 instead of 1.
What values remained the same? Why?
Sine, Cosine, Tangent because the ratios of the sides stay the same.
Why do you think the unit circle is the most commonly used circle in trigonometry?
Your thoughts and observations?