Download Session 14

Session 14
Agenda:
• Questions from 7.3-7.4?
• 8.1 – Angles and their Measure
• 8.2 – The Unit Circle and the Six
Trigonometric Functions
• Things to do before our next meeting.
Questions?
8.1 – Angles and their Measure
• Angles can be given in terms of radians or degrees.
• A full revolution of 360° is equal to 2π radians. Or,
equivalently, 180° is equal to π radians.

• To convert from degrees to radians, multiply by 180 .
• To convert from radians to degrees, multiply by 180 .
• Convert 40° to radians.
• Convert
3
5
radians to degrees.

Common Angles in Degrees and Radians
•
It is important to remember the following common angle
measures in both degrees and radians.
•
Degrees
0
30
45
60
Radians
Degrees
0
90

6

4

3
180
270
360
Radians

2

3
2
2
Quadrants in the Coordinate Plane
•
Positive angles in the coordinate plane are measured from the
positive x-axis in a counterclockwise direction. Negative angles are
measured in the clockwise direction.

2
Quadrant II
or 90
Quadrant I

 or 180
Quadrant III
0
Quadrant IV
3
or 270
2
In what Quadrants do the following angles lie?
240
570
3
4
1000

11
6
10

3
Reference angles
• Given an angle, its reference angle is the smallest angle
formed with the x-axis.
Angle: 60° or

3
Ref. Angle: 60° or

3
Angle: 150° or
5
6
Ref. Angle: 30° or

6
Angle: 225° or
5
4
Ref Angle: 45° or

4
Determine the reference angles for the following.
300
4
3

7
6
8.2 – The Unit Circle and the Six
Trigonometric Functions
• It is very important to
memorize the following
basic sine and cosine
trigonometric values.
• Tangent values do not
necessarily have to be
memorized due to the
identity
sin( )
tan( ) 
cos( )

0  0

6

4

3

2
tan( )
0
3
2
1
2
1
2
1
2
 60
1
2
3
2
 90
0
1
undefined
1
0
0
0
1
undefined
 30
 45
  180
3
2
cos( ) sin( )
1
0
 270
1
3
1
3
•
Consider a circle centered at the origin with radius 1. This is called a
2
2
unit circle and has equation x  y  1
•
Each angle corresponds to a point (x, y) on this unit circle given by
( x, y )  (cos  , sin  )
•
For example, the angle
the unit circle, since

3
corresponds to the point shown below on
  1
cos    ,
3 2
•
3
 
sin   
3 2
Further, since x  cos( ) and y  sin( ) ,
then, for any angle:
cos2 ( )  sin 2 ( )  1
1 3
 ,

2 2 

3
•
The trig values for the angles that
lie on the x and y axes are much
easier to remember by using their
locations on the unit circle and
recalling that the x-coordinate is
the cosine value and the
y-coordinate is the sine value.


2
  0,1
1 3
  ,

3
2 2 

 1 1 
  
,

4
 2 2


 3 1
 
, 
6
2
2


 0
     1, 0
•

(1, 0)
Plotted here are the points on the
unit circle corresponding to the
three main reference angles and
those on the axes.

3
  0,  1
2
•
For angles not in Quadrant I and not on the axes, you can use the
following procedure.
•
Identify the Quadrant the angle is in and its reference angle.
•
The trig values will be the same as those of the reference angle,
with a possible change in sign.
sin( )
x  cos( ) y  sin( ) tan( ) 
cos( )
SINE
ALL
QI
+
+
+
QII
QIII
QIV
•
+
+
-
+
-
The chart to the right indicates which trig
functions are POSITIVE in each quadrant.
TANGENT
COSINE
•
For example, to determine the trig values for  
5
6
, we
know this angle lies in Quadrant II with a reference angle

of 6 . Based on symmetry, the trig values will be the

same as 6 except that in Quadrant II, only sine is
positive.

3
 5 
 
cos 


cos



 
2
 6 
6
 5 
  1
sin 

sin

 
 6 
6 2
 5
tan 
 6
1




3


5
3 1
  
, 
6
2
2


 /6
 /6
 3 1
 
, 
6
2
2


Determine the points on the unit circle corresponding to the
following angles.
7
6
5
3
225
Other Trigonometric Functions
1
cos( )
1
csc( ) 
sin( )
1
cos( )
cot( ) 

tan( ) sin( )
sec( ) 
CSC
ALL
COT
SEC
• The chart displays which of these trig functions are
positive in each quadrant. These can be remembered
based on the definitions and knowing the signs of sine,
cosine, and tangent in each quadrant.
Find all six trigonometric values for the following angles.
240
11
6
23

4
7
2
Trig Functions and Right Triangles
• Given a right triangle with acute angle  , the trig
functions are defined in terms of the sides of the right
triangle:
opp b

hyp c
adj a
cos( ) 

hyp c
opp b
tan( ) 

adj a
sin( ) 
hyp c

opp b
hyp c
sec( ) 

adj a
adj a
cot( ) 

opp b
csc( ) 
b
c

a
• Also recall the Pythagorean Theorem for any right
triangle:
a 2  b2  c2
• In a right triangle with acute angle  , it is known that
cos( ) 
2
. Find all other trig values of
5

.
• Find the length of y in the figure below.
y
45
60
6
•
•
•
Drawing right triangles becomes a very useful technique in
determining trig values of uncommon angles. For example,
suppose you are told that  is in Quadrant I and
4
that cos( )  .
5
We do not know a common angle that has a cosine of 4/5.
Draw a right triangle in Quadrant I that satisfies
the known information and complete the triangle.
All other trig values can then be determined from
this triangle.
5

4
•
In general, the first step is to determine what Quadrant the
angle lies in.
•
If the angle is not in Quadrant I draw a right triangle where the
reference angle has the known information, keeping in mind
that in Quadrants II, III, and IV, signs are different.
3
tan( )  ,  in QIII
2
7
cos( )   ,  in QII
8
8


7
2
7
csc( )   ,  in QIV
5

3
5
7
•
1

and that     , find all other
4
2
trigonometric values for  .
Given that sin( ) 
• Given that sec( x)  
5
and that tan( x)  0 , find all other
2
trigonometric values for x .
• Given that tan( )  9 and that  is not an acute angle, find
all other trigonometric values for  .
Things to Do Before Next Meeting
• Work on Sections 8.1-8.2 until you get all green
bars!!
• Write down any questions you have.
• Continue working on mastering 7.3-7.4. After
you have all green bars on 7.1-7.4, retake the
Chapter 7 Test until you obtain at least 80%.