Download Ch 6 Notes

Pre Calculus
Notes
Chapter 6
Trigonometric Functions
Name______________________________________________#________
Created by W.L. Bass, page 1
Extra Notes
Created by W.L. Bass, page 2
Ms. Bass Pre-Calculus Trig
Chapter 6 Tentative Syllabus
Blue
Day
Blue
Date
Gold
Day
Gold
Date
Tue
Jan 3
Wed
Jan 4
6.1–6.2 Day 1 Notes (Vocab,
Radians, Converting Angles,
Special Triangles)
Day 1 Notes Pop Quiz p.8
Day 2 Notes… p.9 – 10
Thr
Jan 5
Fri
Jan 6
6.1–6.2 Day 2: The Unit Circle
6.1–6.2 Practice
Day 3 Notes… p.13 – 14
Mon
Jan 9
Tue
Jan 10
6.1–6.2 Day 3 Practice
(Working with the Unit Circle)
6.3 Notes… p.17 – 19
Wed
Jan 11
Thr
Jan 12
6.3 Practice: Properties of
Trigonometric Functions
6.4 Notes… p.22 – 24
Fri
Jan 13
Tue
Jan 17
6.4 Practice: Graphing Sine &
Cosine (without phase shifts)
6.5 Notes… p.26 – 28
In-class Topic
Homework
(due next class period)
NO SCHOOL MONDAY JANUARY 16th
Wed
Jan 18
Thr
Jan 19
6.5 Practice: Graphing Secant,
Cosecant, and Tangent
6.6 Notes… p.30 – 32
Fri
Jan 20
Mon
Jan 23
6.6 Practice: Graphing Cotangent
and Sine/Cosine Phase Shifts
Make sure all
practice is complete
Tue
Jan 24
Wed
Jan 25
Put Together Ch 7 Foldable
Chapter 6 Review Day
TBD
Thr
Jan 26
Fri
Jan 27
Chapter 6 Test
7.1 – 7.2 Flipped Notes
Foldable: Back and Top 4
“windows”
Created by W.L. Bass, page 3
Ch 6 Trigonometry
6.1 – 6.2 Day 1 Notes
After completing these notes, you should be able to…








Understand radians v. degrees
Convert between radians and degrees
Draw an angle – positive and negative rotations
Determine what quadrant an angle lies
Determine a reference angle for a given angle
Find co-terminal angles
Find missing measures in special right triangles
Find missing sides in a triangle using the Pythagorean theorem
VOCABULARY!!
An angle is determined by rotating a ray.
The vertex of an angle is the common endpoint.
An angle is in standard position when
the initial side is the positive x-axis.
The initial side of an angle is the side of that
doesn’t move. In standard position, it will be
the positive x-axis.
The terminal side of an angle is the side that moved.
The measure of an angle is determined
by the amount of rotation
A positive angle is generated by a . . .
A negative angle is generated by a . . .
Co-terminal Angles are angles with the same
terminal side. Note: Every angle has an infinite
number of co-terminal angles.
Created by W.L. Bass, page 4
RADIANS… Just another way to measure angles. 
ONE RADIAN:
The portion of the circle (__________________________) that _______________ the radius.
Put it together now…
Half a circle = ____________ radians
and
Half a circle = ____________ degrees
Helpful Mental Images – Show the indicated portions.
2





2
4
3
6
Created by W.L. Bass, page 5
Ex 1
Converting between degrees and radians
a.)
Write 50 in radians.
b.)
Write 3 rad in degrees
8
Ex 2
Sketch, determine its quadrant, find 2 co-terminal angles (positive & negative), find the reference angle
a.)
8
rad
15
Coterm__________ & __________Ref__________
b.)
19
rad
12
Coterm__________ & __________Ref__________
Created by W.L. Bass, page 6
Ex 3
Degrees, Minutes, and Seconds
Conversions…
a.)
1  ________
1 revolution  ________
Convert 70 8'32" to a decimal in degrees
b.)
1'  ________
Convert 38.628 to D M 'S"
Skills You Will Need Next Class
1.
Solve for the missing side in the triangle. (Hint: Use the Pythagorean Theorem.)
a.)
b.)
9
8
4
2.
Special Right Triangle Review…Use your special right triangle patterns to find the missing parts in each
triangle. DO NOT USE PYTHAGOREAN THEOREM.
45- 45- 90 
a.)
30- 60 - 90 
and
b.)
c.)
9
9
14
d.)
e.)
f.)
21
60
30
20
10
60
Created by W.L. Bass, page 7
POP QUIZ/EXIT TICKET… Did you REALLY understand?
Sketch, determine its quadrant, find… the smallest positive & negative co-terminal angles, the reference angle.
1.)
170
Coterm__________ & __________Ref__________
3.)
5
rad
4
Coterm__________ & __________Ref__________
2.)
280
Coterm__________ & __________Ref__________
4.)

7
rad
3
Coterm__________ & __________Ref__________
Created by W.L. Bass, page 8
Ch 6 Trigonometry
6.1 – 6.2 Day 2 Notes
After completing these notes, you should be able to…






Understand and apply arc length formula
Understand and apply area of a sector formula
Understand and apply linear and angular speed
Use a calculator to evaluate trigonometric functions
Understand how the unit circle is developed
Know the points, angles, and signs on the unit circle.
RECALL FROM GEOMETRY
Proof of Arc Length Formula…
 Circumference  Arc Measure 
Arc Length  


1
360



 2 r  Arc Measure 
Arc Length  


360
 1 

  r  Arc Measure 
Arc Length   

180
 1 

 r    Arc Measure 
Arc Length   


1
 1  180 

Thus…
S  r 
Thus…
A
Proof of Area of a Sector Formula…
 Area  Arc Measure 
Area Sector  


360
 1 

2
  r   Arc Measure 
Area Sector  


360

 1 
2
 1   r     Arc Measure 
Area Sector       

180
 2   1   1 

1 2
r 
2
Trigonometric Functions and Their Reciprocals…
sin 
csc  
cos  
sec  

tan 
cot  
Created by W.L. Bass, page 9
Ex 1
Working with arc length
a.)
If the radius of a circle is 10 cm and the arc
length is 18 cm, find the measure of the
central angle in radians.
Ex 2
Working with area of a sector
a.)
Find the area of the sector if the radius of a
circle is 6 cm and the central angle is 150.
Round to hundredths.
Ex 3
Linear speed of an Object Traveling in Circular Motion
b.)
If the radius of a circle is 12 in and the
central angle is 120, find the length of the
intercepted arc.
b.)
If a sector has a central angle of 45 and an
area of 6 cm2 , find the radius of the circle.
Angular Speed (angle traveled over time)


t
Allows us to convert ___________ to ___________
Linear Speed 
dist s r
 
 rw
time t
t
Suppose Greg is spinning a rock at the end of a 3 foot rope at a rate of 150 revolutions per minute (rpm).
a.)
Find the linear speed in feet per minute
when the rock is released.
Ex 4
Use your calculator to evaluate each trigonometric function. Round to 4 decimal places.
a.)
cot  56

b.)
sec  68
b.)

Find the linear speed in miles per hour
when the rock is released.
c.)
csc  29

Created by W.L. Bass, page 10
6.1 – 6.2 Day 2 In-Class Notes
Building the Unit Circle
What do we already know?
45
30
60
Where are we going?
45
30
60
Created by W.L. Bass, page 11
Put It Together – The WHOLE Unit Circle
PRACTICE ROUNDS:
Section 6.1… p.359 – 361
□ Drawing Angles: #11 – 21
□ Converting DMS: #23 – 33
□ Converting Radians/Degrees: #35 – 69
□ Arc Length: #71 – 77
□ Area of a Sector: #79 – 85
□ Applications: #87, 91, 93, 95, 97, 101
Section 6.2… p.375 – 377
□ Using Your Calculator: #65 – 75
Created by W.L. Bass, page 12
Ch 6 Trigonometry
6.1 – 6.2 Day 3 Notes
After completing these notes, you should be able to…





Find trig ratios given one trig ratio
Find trig ratios given a point on a terminal side of an angle
Evaluate trig functions using the unit circle
Evaluate reciprocal trig functions using the unit circle
Understand “All Students Take Calculus” and the domain/range of trig functions
Ex 1
Find the indicated ratios given that  is an acute positive angle…
a.)
Given: sin  
7
11
csc  
cos  
tan 
b.)
Given: cot  
8
5
sin 
sec  
tan 
Ex 2
Let  2, 3 be a point on the terminal side of  . Find the indicated ratios.
csc  
cos  
cot  
Created by W.L. Bass, page 13
USING the Unit Circle
sin 
cos  
tan 
Pythagorean Theorem…
Ex 3
Evaluate each of the following.
a.)
 
sin  
6
b.)
 
cos  
3
c.)
 7 
tan  

 4 
 5 
sec  

 4 
g.)
cos  2 
e.)
 
sin  
2
h.)
 7 
sec 

 6 
i.)
f.)
 
csc   
 2
 5 
csc 

 3 
d.)
Created by W.L. Bass, page 14
6.1 – 6.2 Day 3 In-Class Warm-up
Fill in the key values…
1.)
2.)
3.)
 2 
sin 

 3 
 7 
tan  

 6 
4.)
 4 
csc 

 3 
8.)
 11 
cos 

 6 
5.)
 3 
cot   
 4 
9.)
 
sin   
 4
6.)
 7 
tan  

 3 
10.)
 
sec   
 2
11.)
 13 
tan 

 4 
 3 
cot   
 2 
7.)
sec  4 
Created by W.L. Bass, page 15
12.)
13.)
14.)
 4 
cos 

 3 
15.)
csc  
16.)
 5 
sec  

 4 
 19 
csc 

 6 
 7 
cos 

 4 
17.)
 13 
cot  

 6 
18.)
 7 
csc 

 6 
19.)
 4 
sec  

 3 
20.)
 5 
sin  
 3 
21.)
tan   
PRACTICE ROUNDS:
Section 6.2… p.375 – 377
□
□
□
□
□
Using Points to Evaluate Ratios: #13 – 19
Evaluating Angles: #21 – 45
Finding Ratios (Alternate which ratios you find. Do NOT find all 6 each time.): #47 – 63
Using Points on a Terminal Side: #77 – 83
Extensions: #95 – 111
Created by W.L. Bass, page 16
Ch 6 Trigonometry
6.3 Notes
After completing these notes, you should be able to…




Find trig ratios given one trig ratio and a quadrant restriction
Understand how the Pythagorean Identities are developed
Understand what trig functions are odd/even
Apply odd/even properties to evaluate trig functions
Fill in the key values…
Review: Evaluate each expression.
1.
 
 
sec2    tan 2  
3
4
Ex 1
Finding ratios… two different ways.
a.)
Given:
2.
 is acute & sin  
Find tan 
c).
Given:
   & cos  
Find cot 
12
.
13
2 7
.
7
 
 5 
sin   tan  

6
 3 
3.
 7 
 
sec  
  6 csc  
 4 
3
5
cot   0 & csc    .
3
Find cos 
b.)
Given:
d.)
Given:
sin   0 & tan   1 .
Find sec
Created by W.L. Bass, page 17
Ex 2
Understanding Odd/Even Properties of Trig Functions
a.)
Look at the parent graphs below for each trig function and determine its type of symmetry.
(Rotational around origin or y-axis)
Cosine
Sine
Secant
b.)
Tangent
Cosecant
Cotangent
Earlier we learned that…
even functions have _____________________________ symmetry and f   x   ______________ .
odd functions have _____________________________ symmetry and f   x   ______________ .
Label the functions above as odd or even.
c.)
sin   x   ____________
Thus…
cos   x   ____________
tan   x   ____________
csc   x   ____________
sec   x   ____________
cot   x   ____________
Apply the even/odd properties you just learned to rewrite the expressions below.
sin  2 
 cos  3 
 tan  4 
Created by W.L. Bass, page 18
Ex 3
Understanding Fundamental Identities
a.)
Complete each of the Reciprocal Identities
sin  
1
cos  
1
tan  
1
tan  
csc  
1
sec  
1
cot  
1
cot  
b.)
Developing the Pythagorean Identities
Recall… sin 2 x  cos2 x  1
If we divide the equation by sin 2 x , we get…
If we divide the equation by cos2 x , we get…
sin 2 x  cos2 x  1
sin 2 x  cos2 x  1
These three identities are collectively called the PYTHAGOREAN IDENTITIES.
c.)
Using Fundamental Identities to simplify expressions
1.
sin 2   1
2.
sec2   tan 2 
3.
cot 2   csc2 
4.
tan  
5.
1
 cos 
cos 
6.
cos82 tan 82
sin 82
sec2 
tan 
Created by W.L. Bass, page 19
6.3 In-Class Warm-up
1 – 3… Evaluate each expression.
 
 
1. 2sin    4 cos  
4
6
 5 
2. 4  tan 2  

 3 
 
 
3. 1  cos 2    cos 2  
6
3
4 – 6… Finding each ratio.
12
… find sin 
5
4.
Given:
cos   0 & cot  
5.
Given:
5
tan   0 & sec    … find csc
4
6.
Given: sin  
5
and  is an acute positive angle… Find cos 
12
Created by W.L. Bass, page 20
7 – 12… Simplify each expression
7.
1  csc2 
10.
1  cos2 
11.
tan 20 
8.
sin  cos 

cos  sin 
9.
tan 2   sec2 
12
sin 20
cos 20
1
 
sin 2   
 12  sec2   
 
 12 
PRACTICE ROUNDS:
Section 6.3… p.375 – 377
□
□
□
□
□
□
Applying Co-terminal Angles to Evaluate Angles: #11 – 25
Using “All Students Take Calculus” to Determine Quadrants: #27 – 33
Finding Ratios Given a Ratio
(Alternate which ratios you find. Do NOT find all 6 each time.): #35 – 57
Applying Even/Odd Properties: #59 – 75
Simple Trig Identities: #77 – 87
Extensions: #91, 93
Created by W.L. Bass, page 21
Ch 6 Trigonometry
6.4 Notes
After completing these notes, you should be able to…






Understand all parts of the parent sine curve – Domain, Range, Amplitude, Period, Shape
Understand all parts of the parent cosine curve – Domain, Range, Amplitude, Period, Shape
Identify all key parts (Domain, Range, Amplitude, Period) of a sine/cosine graph without a phase shift.
Graph one period of sine/cosine without a phase shift.
Write the equation for a sine/cosine function without a phase shift given a graph.
Write the equation for a sine/cosine function without a phase shift given the characteristics.
Complete the table…

0




6
4
3
2
2
3
3
4
5
6

7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
sin 
What would it look like as a graph?
Amplitude:__________ Period:______________ Domain:______________ Range:______________
Complete the table…

0




6
4
3
2
2
3
3
4
5
6

7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
cos 
What would it look like as a graph?
Amplitude:__________ Period:______________ Domain:______________ Range:______________
Created by W.L. Bass, page 22
Ex 1
Graph y   sin  2 x 
Vertical Translation:
Amplitude:
Period:
Range:
Distance between each “key value” & calculate “key values”…
Ex 2
Graph y  3  4cos  x 
Vertical Translation:
Amplitude:
Period:
Range:
Distance between each “key value” & calculate “key values”…
Created by W.L. Bass, page 23
y  B  Acos  x  




y  B  Asin  x 
Vertical Translation
Ex 3
a.)
Amplitude
Period




Phase Shift
(Horizontal Translation)
Writing Equations of Sine and Cosine Functions without Phase Shifts… Period  2

Write the equation of a cosine function,
without a phase shift, amplitude = 2,
and period = 4
b.)
Write the equation for the graph below.
Created by W.L. Bass, page 24
6.4 In-Class Warm-up
1.
1 
8 
Graph y  5  2sin  x 
Vertical Translation:
Amplitude:
Period:
Range:
Distance between each “key value” & calculate “key values”…
2.
Write the equation of a sine function…
without a phase shift
x-axis reflection
Amplitude = 4
Period = 1
3.
Write the equation for the graph below.
PRACTICE ROUNDS:
Section 6.4… p.404 – 406
□
□
Finding Period and Amplitude: #11 – 19
Matching Practice: #21 – 30 All
□
□
Graphing: #35 – 57
Writing Equations: #59 – 71
Created by W.L. Bass, page 25
6.5 Notes
Graphs of Tangent and Reciprocal Functions
Ex 1
Graphing Reciprocal Functions
Recall…
 
sec    _______
2
 3
sec 
 2

  _______

csc  0   _______
csc    _______
sec    _______
csc    _______
Graph f  x   5sec  2 x  … Think:________________________________________________________
Created by W.L. Bass, page 26
Ex 2
Graphing Tangent Functions.


2


4
Evaluate tan  for each of the indicated measures.


3

0
4
4
2
5
4
3
2
NOTE: ALL TANGENT GRAPHS WILL INCLUDE TWO PERIODS (i.e. One full revolution)
A.
What does a tangent graph look like?
TANGENT: y  tan x
B.
REFLECTED TANGENT GRAPH:
Graphing Tangent
1
3
Graph… f  x   2 tan  x 
1.
y   tan x


6
Draw a basic curve…
Watch for ______________________ & ______________________
2.
Find the “start” for your graph…
Set equal to _________
3.
Find the period…
Period = 

4.
Find the distance between the key values…
Divide the period into _______ pieces
5.
Calculate the “key values”
Created by W.L. Bass, page 27
y  B  A tan  x  

Vertical Translation
Amplitude

Period
Phase Shift
(Horizontal Translation)
Ex 3 f  x   4  tan  3x 
Reflection or Vertical Translation?
Starting point:
Period:
Distance b/w key values:
Calculate the key values:
Domain:__________________________________
Created by W.L. Bass, page 28
6.5 In-Class Warm-up
1.
Graph f  x   2  3csc  x  … Think:_____________________________________________________
2.
Graph f  x   1  3 tan  x 




4
Reflection or Vertical Translation?
Starting point:
Period:
Distance b/w key values:
Calculate the key values:
Domain:__________________________________
PRACTICE ROUNDS:
Section 6.5… p.414
□
□
Basic Skills: #7 – 15
□
Extensions: #41, 43, 45, 47, 49
Graphing: #17 – 39
Created by W.L. Bass, page 29
6.6 Notes
Graphs of Cotangent and Phase Shifts for Sine and Cosine
Ex 1
Graphing Cotangent Functions
 
tan    _______
2
Recall…
 3
tan 
 2

  _______

cot  0   _______
cot    _______
tan    _______
cot    _______


Graph f  x   2cot  x 


3
Think:
Reflection or Vertical Translation?
Starting point:
Period:
Distance b/w key values:
Calculate the key values:
Domain:__________________________________
Created by W.L. Bass, page 30
Phase Shift (Horizontal Translation)
When a sine or cosine graph has a phase shift (horizontal translation),
you must first determine your starting point for the graph!
To find the starting point, solve…  x    0
Ex 2


Graph: f  x    sin  x  
4

Vertical Translation:
Amplitude:
Starting point…
Period:
Distance between each “key value”
Calculate “key values”…
Domain:____________________Range:__________________
Created by W.L. Bass, page 31
Ex 3
Graph: f  x   cos  4 x  2 
Vertical Translation:
Amplitude:
Reflection?
Starting point…
Period:
Distance between each “key value”
Calculate “key values”…
Domain:____________________Range:__________________
Ex 4


Graph: f  x   csc  2 x   … Think:___________________________________________________
2

Vertical Translation:
Amplitude:
Reflection?
Starting point…
Period:
Distance between each “key value”
Calculate “key values”…
Domain:____________________Range:__________________
Created by W.L. Bass, page 32
6.6 In-Class Warm-up
1.
1

x    … Think:_____________________________________________
2

Graph… f  x   1  2sec 
Vertical Translation:
Amplitude:
Reflection?
Starting point…
Period:
Distance between each “key value”
Calculate “key values”…
Domain:____________________Range:__________________
2.
Graph… f  x   3  cot  4 x  … Think:_____________________________________________
Reflection or Vertical Translation?
Starting point:
Period:
Distance b/w key values:
Calculate the key values:
Domain:__________________________________Range:__________________
Created by W.L. Bass, page 33
3.
Writing Equations with Phase Shifts
Recall… y  B  A sin  x    and Period 
Thus…     Phase Shift 
2

Why?
a.)
Sine
Amplitude: 3
Period:
b.)

2
Phase Shift: 2
Cosine – Reflected across x-axis
Amplitude: 5
Period: 3
1
Phase Shift:
3
PRACTICE ROUNDS:
Section 6.6… p.424
□
□
Graphing Sine/Cosine Phase Shift: #3 – 13
Writing Equations: #15 – 18 All
□
Mixed Graphing Review: #19 – 25
(Tangent, Cotangent, Secant, Cosecant)
Created by W.L. Bass, page 34