(Sections 5.4 and 5.5: More Trig Identities) 5.42
SECTIONS 5.4 and 5.5: MORE TRIG IDENTITIES
PART A: A GUIDE TO THE HANDOUT
See the Handout on my website.
The identities (IDs) may be derived according to this flowchart:
In Calculus: The Double-Angle and P
(Section 5.3: Solving Trig Equations) 5.32
Third factor
u 1= 0
u =1
sin x = 1
x = + 2 n
2
( n integer )
Solution set:
5
+ 2 n, or x = + 2 n
x x = n, x = + 2 n, x =
6
6
2
( n integer )
When gathering groups of solutions, you should check to see if there
a
(Section 5.3: Solving Trig Equations) 5.22
SECTION 5.3: SOLVING TRIG EQUATIONS
PART A: BASIC EQUATIONS IN sin, cos, csc, OR sec (LINEAR FORMS)
Example
Solve: 5 cos x 2 = 3 cos x
(It is assumed that you are to give all real solutions and to give them in ex
(Section 5.2: Verifying Trig Identities) 5.10
SECTION 5.2: VERIFYING TRIG IDENTITIES
PART A: EXAMPLE; STRATEGIES AND SHOWING WORK
One Example; Three Solutions
Verify the identity:
csc + cot
= cot csc .
tan + sin
Strategies and Showing Work
To verify an
(Answers for Chapter 0: Preliminary Topics) A.0.1
CHAPTER 0: Preliminary Topics
SECTION 0.1: SETS OF NUMBERS
1)
12 is an integer. 12 ,
are real numbers.
5 , and 7.13 are rational numbers. All the listed numbers 7
2) 3)
12 is an integer, or 12 is in the se
(Section 5.1: Fundamental Trig Identities) 5.01
CHAPTER 5: ANALYTIC TRIG
SECTION 5.1: FUNDAMENTAL TRIG IDENTITIES
PART A: WHAT IS AN IDENTITY?
An identity is an equation that is true for all real values of the variable(s) for which all
expressions contain
(Section 4.8: Applications) 4.90
SECTION 4.8: APPLICATIONS
PART A: WORD PROBLEMS
We did a word problem in Notes 4.32. Some tips:
Read the problem carefully. Underline, highlight, or summarize key pieces of
information so that you dont have to keep reread
(Section 4.7: Inverse Trig Functions) 4.72
SECTION 4.7: INVERSE TRIG FUNCTIONS
You may want to review Section 1.8 on inverse functions.
PART A : GRAPH OF sin-1 x (or arcsin x )
Warning: Remember that f
(or reciprocal). Usually, f
1
1
denotes function inve
(Section 4.7: Inverse Trig Functions) 4.72
SECTION 4.7: INVERSE TRIG FUNCTIONS
You may want to review Section 1.8 on inverse functions.
PART A : GRAPH OF sin-1 x (or arcsin x )
Warning: Remember that f
(or reciprocal). Usually, f
1
1
denotes function inve
(Section 4.6: Graphs of Other Trig Functions) 4.63
Example
Use the Frame Method to graph one cycle of the graph of
2
y = 2 tan x 3 . (There are infinitely many possible cycles.)
5
Solution
2
> 0 . If b < 0 , we would need to use the Even/Odd
5
Propertie
(Section 4.6: Graphs of Other Trig Functions) 4.53
SECTION 4.6: GRAPHS OF OTHER TRIG FUNCTIONS
()
PART A : GRAPH f = tan
We begin by tracing the slope of the terminal side of the standard angle as increases
from 0 towards
and as it decreases from 0 towar
4.41
PART E: PERIOD
()
()
()
()
We now consider the forms f x = a sin bx and f x = a cos bx .
In Sections 4.5 and 4.6, we assume that b is a positive real number.
Accordion Effects
Recall from Section 1.6: Notes 1.62 that, if x replaced by bx, then the
co
4.33
SECTION 4.5: GRAPHS OF SINE AND COSINE FUNCTIONS
()
PART A : GRAPH f = sin
()
Note: We will use and f for now, because we would like to reserve x and y for
discussions regarding the Unit Circle.
We use radian measure (i.e., real numbers) when we gra
4.21
PART C: EXTENDING FROM QUADRANT I TO OTHER QUADRANTS
Reference angles
The reference angle for a non-quadrantal standard angle is the acute angle that its
terminal side makes with the x-axis.
Brothers (the authors term) are angles that have the same r
4.08
SECTIONS 4.2-4.4: TRIG FUNCTIONS
(VALUES AND IDENTITIES)
We will consider two general approaches: the Right Triangle approach, and the Unit Circle
approach.
PART A: THE RIGHT TRIANGLE APPROACH
The Setup
The acute angles of a right triangle are comple
4.01
CHAPTER 4:
TRIGONOMETRY (INTRO)
SECTION 4.1: (ANGLES); RADIAN AND DEGREE MEASURE
PART A: ANGLES
An angle is determined by rotating a ray (a half-line) from an initial side to a
terminal side about its endpoint, called the vertex.
A positive angle is
3.27
SECTIONS 3.4 AND 3.5:
EXPONENTIAL AND LOG EQUATIONS AND MODELS
PART A: ONE-TO-ONE PROPERTIES
(Assume that b is nice.)
The bx and log b x functions are one-to-one. (Their graphs pass the HLT.) Therefore,
(b
1)
2)
( log
b
M
= bN
M = log b N
)
)
( M = N
3.18
SECTION 3.3: (MORE) PROPERTIES OF LOGS
PART A: READING LOG EXPRESSIONS
We will use grouping symbols as a means of clarifying the order of operations in
expressions.
Often, grouping symbols are omitted when they could have helped.
How do we read log e
3.13
SECTION 3.2: LOGARITHMIC (LOG) FUNCTIONS AND
THEIR GRAPHS
PART A: LOGS ARE EXPONENTS
Example
Evaluate: log 3 9
Solution
The question we ask is: 3 to what exponent gives us 9?
log 3 9 = 2 , because
logarithmic form
exponential form
We say: Log base
3.01
CHAPTER 3:
EXPONENTIAL AND LOG FUNCTIONS
SECTION 3.1: EXPONENTIAL FUNCTIONS AND
THEIR GRAPHS
PART A: THE LEGEND OF THE CHESSBOARD
The original story takes place in the Middle Ages and involves grains of wheat.
Instead, we shall transport ourselves to
(Section 2.7: Nonlinear Inequalities) 2.77
SECTION 2.7: NONLINEAR INEQUALITIES
We solved linear inequalities to find domains, and we discussed intervals in Section 1.4:
Notes 1.24 to 1.30.
In this section, we will solve nonlinear inequalities to find doma
2.66
SECTION 2.6: RATIONAL FUNCTIONS
PART A: ASSUMPTIONS
()
()
Assume f x is rational and written in the form f x =
()
()
()
( ),
D ( x)
Nx
where N x and D x are polynomials, and D x 0 (i.e., the zero polynomial).
()
()
Assume for now that N x and D x hav
THE QF METHOD FOR FACTORING QUADRATICS
()
Remember our old friend f x = 4 x 3 5x 2 7 x + 2 .
Lets say we want to factor this completely over C.
From Part B on the Rational Zero Test, we found a list of candidates for rational zeros.
It turned out that 2 w
2.57
PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA)
The Fundamental Theorem of Algebra (FTA)
()
If f x is a nonconstant n th -degree polynomial in standard form with real
coefficients, then it must have at least one complex (possibly real) zero.
Put Ano
2.46
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
()
Assume f x is a nonconstant polynomial with real coefficients written in standard form.
PART A: TECHNIQUES WE HAVE ALREADY SEEN
Refer to:
Notes 1.31 to 1.35
Section A.5 in the book
Notes 2.45
Refe
2.35
SECTION 2.4: COMPLEX NUMBERS
Let a, b, c, and d represent real numbers.
PART A: COMPLEX NUMBERS
i, the Imaginary Unit
We define: i = 1 .
i 2 = 1
(
)
If c is a positive real number c R + , then
c = i c.
Note: We often prefer writing i c , as opposed t
2.25
SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL
DIVISION
PART A: LONG DIVISION
Ancient Example with Integers
4
2
9
8
We can say:
1
9
1
= 2+
4
4
By multiplying both sides by 4, this can be rewritten as:
9 = 42 +1
In general:
dividend, f
divisor, d
=
(
quot
2.10
SECTION 2.2: POLYNOMIAL FUNCTIONS OF
HIGHER DEGREE
PART A: INFINITY
The Harper Collins Dictionary of Mathematics defines infinity, denoted by , as a
value greater than any computable value. The term value may be questionable!
Likewise, negative infin
2.01
CHAPTER 2:
POLYNOMIAL AND RATIONAL FUNCTIONS
SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS)
PART A: BASICS
If a, b, and c are real numbers, then the graph of
f x = ax 2 + bx + c is a parabola, provided a 0 .
()
=y
If a > 0 , it opens upward.
If a <
(Section 1.9: Inverse Functions) 1.99
SECTION 1.9: INVERSE FUNCTIONS
PART A: ONE-TO-ONE (1-1) FUNCTIONS; HORIZONTAL LINE TEST (HLT)
A function is one-to-one (1-1) No two inputs from its domain yield the same
output.
Remember that a function by definition