Table of Contents
1. Angles and their Measures
Angles and their Measures
Essential question – What is the
vocabulary we will need for
trigonometry?
Make a table
• Term
Definition
Picture
Trigonometry vocabulary
Initial side – start side of angle
Terminal side – end side of angle
Standard position – An angle whose initial
side is on the positive x-axis
Positive angles
• An angle in standard position that rotates
counterclockwise
Negative angles
• An angle in standard position that rotates
clockwise
Coterminal Angles
• Angles that have the same terminal side
Quadrants
Quadrant II
Quadrant III
Quadrant I
Quadrant IV
Angles of the axes
900 , 2700
1800 , 1800
00 ,3600
2700 , 900
• Variables you will see for angle measures
  theta
  alpha
  beta
Radians
• Angle measures can also be expressed in
radians
• A radian is the ratio of the length of an arc to its
radius
• Radians are expressed in terms of
o
•
= 180

• To change from degrees to radians, multiply by180
and reduce.
180
• To change from radians to degrees, multiply by 


Radians continued
• Radians can take 2 forms – an exact answer
and an approximate decimal answer
• The exact answer has a  in it and it is the usual
way to see radians
• To find an exact answer with your calculator, do
not put the  in the calculator, only write it in the
answer
• However, radians can also be written as a
decimal without the 
Angles of the axes

2

0, 2
3
2
Examples
• Change from degrees to radians
36o
 250o
360o
• Change from radians to degrees
4
3
16
Coterminal angles
• You add or subtract multiples of 360o (or 2π)
to find coterminal angles
• Find 2 coterminal angles (one positive and one
negative) for 35o
• Find 2 coterminal angles (one positive and one
negative) for -23o
• Find 2 coterminal angles (one positive and one
negative) for 740o
Examples for radians
• Find a positive and negative coterminal
angle
 4
3
16
What quadrant is it in?
• To find out what quadrant an angle is in
– Make a negative angle positive by adding 360o or 2π
(may need to do multiple times)
– If angle is bigger than 360o or 2π, make it smaller by
subtracting 360o or 2π (may need to do multiple
times)
– Figure out what quadrant it is in based on angles of
axes (from yesterday)
– If the question asks you to sketch the angle,
• draw the terminal side in the right quadrant
• go in either positive or negative direction based on original
problem
• if you have added or subtracted 360o or 2π, you need to go
around multiple times.
What quadrant is it in (and
sketch)?
332o
156o
1000o
240o
What quadrant is it in? (radians)
•
•
•
•
Follow steps to make small positive angle
Put fraction in calculator (without the π)
If answer is < 0.5, it is in 1st quadrant
If answer is between 0.5 and 1, it is in 2nd
quadrant
• If answer is between 1 and 1.5, it is in 3rd
quadrant
• If answer is between 1.5 and 2, it is in 4th
quadrant
Examples – which quadrant?
(radians) (and sketch)
7
5
3

16
34
5
Reference Angles
• A reference angle is the acute angle that
an angle makes with the x-axis
•
Finding Reference Angles
• Follow steps to make small positive angle
• Find out which quadrant it is in
• In the 1st quadrant, the reference angle is the
SAME as the angle itself
• In the 2nd quadrant subtract the angle from
180o or π
• In the 3rd quadrant subtract 180o or π from
the angle
• In the 4th quadrant subtract the angle from
360o or 2π
Examples
•
•
•
•
•
•
•
Find the reference angle for the following angles.
37o
7π/4
-2π/3
-190o
17π/7
820o
Assessment
• 321
– Write 3 new things you learned
– Write 2 vocabulary words with their meaning
– Write 1 thing you don’t understand