AAT – Unit 3
Angles and The Unit Circle
Section
Topic
Flying Into Trig on a Paper Plate
5-1
AAT-15
Angles and Radian Measure
Converting from degrees to
radians and vice versa
Coterminal angles
5-4 and
Supplement
AAT-13,14
The Unit Circle
Trig Functions of Real Numbers
Assignment
Memorize The
Paper Plate
Practice Page 8
p. 480:14-40 even
Quiz on 5-1
Page 11
Page 13
Quiz on Unit Circle
p. 582-583:
2-16 even, 43-58
Review for Test,
pages 15 - 16
Test on 5-1 and
5-4
Due:
2
Flying into Trig on a Paper Plate
Warm-up:
1. Label the quadrants:
2. Classify the following angles as obtuse, acute or right:
a) 34°
b) 91°
c) 90°
d) 128°
3. Add the following fractions (without a calculator!)
3 5
2 1


a)
b)
8 2
3 4
Flying Into Trig on a Paper Plate - Day 1
Intro:
1. Locate and mark the center of the circle. (Fold the paper plate in half and then in half
again). We will call the center point O. Mark your center point with the letter O.
The unit circle is a circle with a radius equal to one and the center at the origin of a
rectangular coordinate system.
2.
What is the circumference of the above circle? ________________________
3.
Darken (mark) the folds and label the x and y axis with a colored marker. Do not use the
black marker.
4.
Locate the point where the positive x-axis intersects the unit circle and label it A using the
black marker.
This represents the anchor point (the beginning)
5.
Pinch the anchor point with one hand and with the other hand trace an arc with length of
2π.
How many times around the circle is 2π? ________
How many times around the circle is 6π? ________
How many times around the circle is 24π? ________
How many times around the circle is π? ________
3
Activity 1
ALWAYS check with the instructor BEFORE putting anything on your paper plate.
Note: when you trace arcs, you always go counterclockwise
1.
If you trace the length of the arc from A (the anchor point) all the way around the UNIT
CIRCLE back to A, what is the length of the arc traced? __________
2.
If you trace the length of the arc from A to where the positive y-axis intercepts the UNIT
CIRCLE, what is the length of the arc traced? __________
If you trace the length of the arc from A to where the negative x-axis intercepts the
UNIT CIRCLE, what is the length of the arc traced? __________
3.
4.
If you trace the length of the arc from A to where the negative y-axis intercepts the
UNIT CIRCLE, what is the length of the arc traced? __________
5.
What is the length of OA ? __________
6.
What do you think is the “initial point” of ALL arcs above? __________
7.
How would you describe the “terminal point” of an arc? __________________
8.
After completing questions 1 – 7 you are now ready to label your unit circle. The following
steps will tell you how to mark your plate. Make sure you know what to label and how
before you start.
9.
We will be labeling the quadrantal arcs, so you will need to decide on 4 colors to use, not
black.
10. First, with one color, starting at A (the anchor point), trace along the very edge of the
paper plate until you reach 2π. (one trip around the circle). Label 2π.
11. Second, now using a second color starting at A, trace along the edge of the paper plate until
you reach
3
3
(on the negative y-axis) Label
.
2
2
12. Third, using a third color starting at A, trace along the edge of the paper plate until you
reach π ( the negative x-axis) Label π.
13. Lastly, using a fourth color starting at A, trace along the edge of the paper plate until you
reach

2
. (The positive y-axis) Label

2
.
Definition of Radian Measure:
Angles in Trigonometry are measured in ______________ and ______________.
180  _______________radians
4
Fill in the following table, using your paper plate:
Arc Measurement

Angle, in Radians
Angle , in Degrees
2

3
2
2
1. Using the above table, how would you convert an angle from radians to degrees?
2. How would you convert from degrees to radians?
Example Convert from degrees to radians:
a) 200°
b) 34°
c) 90°
d) 128°
Example Convert from radians to degrees:
a).
12
7
b). 
24
5
c). 17.46 radian
Activity Two
1. Fold the plate along the x-axis then fold it along the y-axis (fourths). Now fold the plate
again so that the plate is divided into 8 equal parts. Unfold the plate.
2. A new “family rectangle” has formed with ________ new values for their arcs.
3. List the values for the new arc lengths in quadrants I, II, III, and IV in terms of
a. __________
__________
__________
4. Label your plate with these new radian measures.
__________
.
5
Activity Three
ALWAYS check with the instructor BEFORE putting anything on your paper plate.
1.
Fold the paper plate along the x-axis and then fold the remaining half plate in thirds.
Unfold the plate.
2.
A new “family rectangle” has formed with ________ new values for their arcs.
3.
List the values for the new arc lengths in quadrants I, II, III, and IV in terms of
__________
4.
__________
__________
.
__________
Label your plate with these new radian measures.
Activity Four
ALWAYS check with the instructor BEFORE putting anything on your paper plate.
1.
Fold your plate along the x-axis and then fold the remaining half plate into thirds again.
Now fold it in half.
2.
How man pieces have we folded the plate into? __________
3.
A new “family rectangle” has formed with ________ new values for their arcs.
4.
List the values for the new arc lengths in quadrants I, II, III, and IV in terms of
__________
5.
__________
__________
__________
Label your plate with these new radian measures.
Activity Five
Label all radian measures with the corresponding degree measures.
.
6
Angles in Standard Position
Examples: Draw the following angles in standard position:
a) 𝜃 =
𝜋
4
b) 𝛼 =
5𝜋
6
c) 𝛽 = −
2𝜋
3
d) 𝛾 =
7𝜋
3
Coterminal Angles are angles in standard position (angles with the initial side on the positive xaxis) that have a common terminal side. For example 30°, –330° and 390° are all coterminal.
Examples: Name a coterminal angle to each of the following angles.
a) 𝜃 =
𝜋
4
b) 𝛼 =
5𝜋
6
c) 𝛽 = −
2𝜋
3
d) 𝛾 =
7𝜋
3
7
Reference Angles: The reference angle is the acute angle formed by the terminal side of the
given angle and the x-axis. Reference angles may appear in all four quadrants. Angles in
quadrant I are their own reference angles.
Remember: The reference angle is measured from the terminal side of the original angle
"to" the x-axis (not the y-axis).
Examples: Name the reference angle:
a) 𝜃 =
𝜋
4
b) 𝛼 =
5𝜋
6
c) 𝛽 = −
2𝜋
3
d) 𝛾 =
7𝜋
3
8
Practice
State the quadrant in which the terminal side of the angle lies.
1. 310°
2. -100°
3. 180°
4. 23°
Find a co-terminal angle that is between zero and 360° or 2 .
3. 657
2. 311
1. 832
4.

5. 
6

6.
3
5
6
If each angle has the given measure and is in standard position, determine the quadrant in which
each terminal side lies. Name the co-terminal angle.
7. 
8
3
8. 167
9.
7
4
11. 
10. 635
4
3
12. 730
Find one positive and one negative co-terminal angle.
13. 60
14.
11
6
Find the reference angle for each given angle.
15. 30
19. 
9
4
16.
5
4
17. 130
20.
23
6
21. 
5
3
18. 210
21. 420
9
Flying into Trig on a Paper Plate: Creating the Unit Circle
Warm-UP: Using the paper plate you have created:
1. What is the length of OA ? __________
2. What are the coordinates of the point where:
3. the positive x-axis intercepts the UNIT CIRCLE? __________
4. the positive y-axis intercepts the UNIT CIRCLE? __________
5. the negative x-axis intercepts the UNIT CIRCLE? __________
6. the negative y-axis intercepts the UNIT CIRCLE? __________
Label these coordinates on your paper plate.
Activity One
ALWAYS check with the instructor BEFORE putting anything on your paper plate.
1.
Remembering the special right triangles from Geometry, we can find the x and y coordinate
of each corresponding value on the “family rectangle”.
2.
In the isosceles right triangle, remember that the length of the sides have the ratio
1:1: 2
3.
(Note: order is x: y: hypotenuse)
However, since we are using the UNIT CIRCLE, our hypotenuse is 1. So we will divide each
of the three values by
2 in order to maintain the proper ratio and get the hypotenuse
to be 1.
4.
The new ratio values are __________:__________:__________.
5.
We also must remember to rationalize the denominators, so the new values are
__________:__________:__________.
6.
Using the ratio of an Isosceles Triangle, complete the following triangle. Label the arc
measures, the sides of the triangle, and the coordinates of the arc measures.
7.
Label the coordinates on your paper plate.
10
Activity Two
1.
The ratio of the lengths of the sides in a 30˚: 60˚: 90˚ triangle is
3 :1: 2 (Note: order is x:
y: hypotenuse)
2.
However, since we are using the UNIT CIRCLE, our hypotenuse is 1. So we will divide each
of the three values by ____ in order to maintain the proper ratio and get the hypotenuse
to be 1.
3.
The new ratio values are __________:__________:__________.
4.
Complete the following triangle. Label the arc measures, the sides of the triangle, and the
coordinates of the arc measures.
5.
Label the coordinates on your paper plate.
Activity Four
ALWAYS check with the instructor BEFORE putting anything on your paper plate.
1.
The ratio of the lengths of the sides in a 60˚: 30˚: 90˚ triangle is 1: 3 : 2 (Note: order
is x: y: hypotenuse)
2.
However, since we are using the UNIT CIRCLE, our hypotenuse is 1. So we will divide each
of the three values by ____ in order to maintain the proper ratio and get the hypotenuse
to be 1.
3.
The new ratio values are __________:__________:__________.
4.
Complete the following triangle. Label the arc measures, the sides of the triangle, and the
coordinates of the arc measures.
5.
Label the coordinates on your paper plate.
11
Homework - Use the following diagram to complete the unit circle. Include radian measures,
degree measures and coordinates.
12
Warm-up: Simplify:
2
2
1.
2
2
2
2
3.
5.
3
2
1
2
2.
2
3
4.
1
1/ 2
6.
1
2
3
2
Remember for every point on the unit circle, we can draw a right triangle:
Using Right Triangle Trig, write down the following in terms of x and y:
sin   __________
cos  __________
tan   __________
Use your paper plate to find the exact values of the following trig functions:

1. cos 45
2. sin
5. sin 0
6. cos90
9. tan

3
4
10. tan
5
6
2
3
3.
cos
4. sin
7.
sin 270



 
11. tan  


2 
5
2
8. cos
12. tan 0
7
6
13
Homework – Use your Unit Circle to find the following:
5
4
1. sin 45
2. cos
5. cos0
6. sin 90
9. tan
2
3

10. tan
7
6

4
3
3.
sin
7.
cos60
11. tan
3
2
4. cos

2

8. sin  

12. tan
 

6 

4
Go online to research answers to the following questions:
1. So far we have used three trig functions – sine, cosine and tangent. But actually there are
six trig functions! What are they?
2. What are the reciprocal identities? List them here:
14
Warm-Up: From Last notes….
Remember for every point on the unit circle, we can draw a right triangle:
Using Right Triangle Trig, write down the following in terms of x and y:
sin   __________
csc  __________
cos  __________
sec  __________
tan   __________
cot   __________
Find each exact value using the Unit Circle.

1.
Sin
5.
Sin 300
9.
Sin
13.
Sec
17.
Sec 270
4
8
3
9
2


3.
Tan
5
3
4.
Cos 210
Tan 330
7.
Sin
3
4
8.
Cos 240
10.
Cos 420
11.
Tan 
12.
Csc 330
14.
Cot 135
15.
Cot  5
16.
19.
Cos  20
20.
2.
Cos
6.
18.
5
6

Tan 210

Csc 540
Sin
13
4
15
Test Review
If each angle has the given measure and is in standard position, determine the quadrant in which
each terminal side lies.
1.
7
12
6. 1000°
2.

2
3
3. 371°
7.

23
4
8.
4
5
4.
17
8
9.

5. –156°
29
13
10. –240°
Change each degree measure to radian measure.
11. 36°
12. –250°
13. 345°
Change each radian measure to degree measure.
14. –1
3
16
15.
16.

7
9
17. –2.56
Find one positive and one negative angle that are co-terminal with the given angle.
18. 70°
19.

2
5
20. –300°
21.
3
4
Find the reference angle for each angle with the given measure.
22. –20°
23.
160°
24. –545°
25.
7
9
26.

7
3
16
Refer to the unit circle to help you evaluate the following trigonometric functions.

27. Sec 300˚
28. Cos 210˚
29. Cotan (–225˚)
30. Csc
31. sin 2
32. Tan 
33. Sec 11
34. Tan (–150˚)
3
4
35. Cos 2
36. Tan (–300˚)
3
39. Sec 
40. Cos 5
2
3
6
37. Cotan 7
4
38. sin (–30˚)