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TRIGONOMETRY NOTES
By
STEVEN SY
Copyright 2008
2
Contents
0
...
8
0
...
9
1 Review of Functions
15
1
...
15
1
...
23
1
...
29
1
...
35
1
...
42
1
...
47
1
...
51
1
...
54
1
...
59
3
1
...
62
1
...
67
1
...
73
1
...
75
1
...
78
2 Rational Functions
87
2
...
87
2
...
89
2
...
96
3 Elementary Trigonometry
111
3
...
111
3
...
118
3
...
4
The Wrapping Function At Multiples of
...
5
The Wrapping Function At Multiples of and
...
6
The Trigonometric Functions: Definitions
...
7
Domain and Range of the Trigonometric Functions
...
124
3
...
159
3
...
161
3
...
165
4 Graphing Trigonometric Functions
173
4
...
173
4
...
179
4
...
186
4
...
189
4
...
193
4
...
199
4
...
205
4
...
210
5 Trigonometric Identities
217
5
...
217
5
...
223
5
...
233
5
...
244
5
5
...
248
5
...
252
5
...
257
5
...
262
5
...
266
5
...
270
5
...
274
6 Advanced Trigonometric Concepts
279
6
...
279
6
...
285
6
...
289
6
...
298
6
...
307
6
...
319
6
...
329
6
...
332
6
...
346
6
6
...
355
7 Triangle Trigonometry
363
7
...
363
7
...
375
7
...
387
7
...
391
7
...
398
7
...
403
7
...
415
Selected Answers to the Exercises
421
7
0
...
DO NOT BLINDLY APPLY powers and roots across expressions that have
2
...
As in comment 1,
or
signs
...
The square formula applies:
...
This means when you
square you will have a term that looks like twice the product of the terms in parentheses
...
...
Note that you can
4
...
8
0
...
Formulas
Perfect Square Factoring:
Difference of Squares:
Difference and Sum of Cubes:
B
...
There is no “sum of squares” formula, i
...
no formula for
(over the real numbers)
...
With and in the same equation, you get one equation when you take the “top” signs,
and you get another when you take the “bottom” signs
...
3
...
Then the second factor looks
like you “square” the first factor, except rather than doubling the middle term, you take the
negative of the middle term
...
Examples
Example 1:
Factor
Solution
Now use the Perfect Square Formula (with minus):
Ans
Example 2:
Factor
Solution
Now use the Difference of Squares Formula:
Ans
10
Example 3:
Factor
Solution
Now use the Sum of Cubes Formula:
Ans
Example 4:
Factor
Solution
Now use the Difference of Cubes Formula:
Ans
11
Exercises
1
...
Expand
3
...
Factor
5
...
Factor
7
...
...
...
...
12
13
14
Chapter 1
Review of Functions
1
...
Definition of a Function
Every valid input, , produces exactly one output, ; no more, no less
B
...
Implicit Functions
1
...
2
...
15
C
...
Example 1:
Solution
Plug in some -values, see how many -values you get:
For each , we only get one y, so this a function of
...
Example 2:
Solution
...
So this is not a function of
...
16
Example 3:
Solution
...
For each , we only get one y, so this a function of
...
Note: If we have a graph, we may determine if we have a function of by using the
Vertical
Line Test (if any vertical line hits the graph more than once, it is not a function
of )
...
Notation and Comments
1
...
2
...
input, is the output ( -value)
...
In terms of ,
is a -value
...
18
to spit back
For instance, if 3 is an
E
...
Use formula for the output,
, so
, and plug in
...
21
Example 2:
Find
and simplify
Solution
We wantthe output when the input is
output where you see :
Now
, so
Then simplify:
Ans
...
Example 3:
Simplify the difference quotient
Solution
Now
so
Then
Ans
, if
20
...
Determine if the following equations define functions of ; if so, state whether they are explicit
or implicit
...
b
...
2
...
g
...
d
...
Find and simplify:
a
...
c
...
e
...
Note: This is different than part c
...
Let
a
...
c
...
Find and simplify:
d
...
Find the difference quotient
5
...
Find
and simplify for
and simplify for
and simplify for
...
22
...
2 Domain and Range of Functions
A
...
Range
all outputs
C
...
In particular, we don’t allow division by zero or complex numbers
...
Denominators: Throw away values making the denominator zero
...
Even Roots: Set inside
, and solve inequality
...
Logarithms: Set inside
, and solve inequality
...
Domain Finding Examples
Example 1:
Given
, find
...
Ans
Example 2:
Given
, find
...
Denominator: Throw away
2
...
Throw away
Ans
24
E
...
Methods
1
...
What
is the smallest -value? What is the largest -value? Are any -values missed?
Heuristic: Expressions that are raised to even powers or even roots of expressions have smallest -value equal to 0
...
Graph it, and read off the -values from the graph
...
See if you can apply “HSRV transformations” to a known base graph (reviewed
later in Section 1
...
For a quadratic function, find the vertex
...
5
...
There are other methods, such as the Back Door method, which will not be reviewed
here
...
Range Finding Examples
Example 1:
Given
, find
...
What is the largest -value possible? There is no upper limit!
(The -values go to
...
Ans
Example 2:
Given
, find
...
)
largest -value possible? no upper limit
no values larger than
Ans
are missed
26
Example 3:
Given
, find
...
The parabola opens down since the leading coefficient
is negative
...
Vertex Formula:
largest -value is
Ans
27
!
...
Find
2
...
b
...
for where
a
...
c
...
e
...
3
...
Every polynomial
satisfies
b
...
No quadratic function has
4
...
3 HSRV Transformations
A
...
Horizontal Translation: Add/Subtract Number INSIDE of (Left/Right Respectively)
a
...
-values change
2
...
-values fixed
b
...
Reflections:
sign:
a
...
Inside
-axis reflection, -values fixed, -values times
sign (next to
): -axis reflection, -values fixed, -values times
4
...
-values fixed
b
...
NOTE: When the output formula is not given, we identify “key points” on the graph, and
then move those according to the rules given
...
B
...
Also determine
30
and
...
H: Add 1 inside: move graph left by 1; -values fixed, -values move
2
...
R: Outside
sign:
-axis reflection; -values fixed, -values times
R
4
...
Let be given by the following graph:
a
...
Graph , where
c
...
Graph , where
e
...
Graph , where
33
2
...
Then determine
...
Graph , where
b
...
Suppose is a function with
Find
and , where
and
34
...
and
...
1
...
Symmetry of Functions
1
...
2
...
Note: A nonzero function may not have -axis symmetry
...
Even/Odd Tests
1
...
2
...
To use these tests, we compute and and compare to
...
If none of them are equal, the function is
neither even nor odd
...
For example, if
use generic x, not
When testing,
, then and , but is not even!
C
...
36
Example 2:
Solution
Ans
Since
Example 3:
,
is even
...
D
...
Examples of symmetric domains:
,
,
,
Examples of domains that are not symmetric:
,
,
,
E
...
Statement: Any function (even/odd/neither) with a symmetric domain can be decomposed
into the sum of an even function and an odd function
...
Formulas:
a
...
odd
3
...
Note, by definition,
b
...
odd
38
odd
is an odd function
...
Decomposition Examples
Example 1:
, decompose
Given
into even and odd parts
...
Now use the formulas:
even
odd
Ans
Note:
even
odd
even
odd
...
Given
Solution
Note:
, so the domain is symmetric
...
was already odd
...
To test a function, do the decomposition
...
If the odd part is 0, the function is even
...
40
Exercises
1
...
b
...
d
...
f
...
2
...
3
...
4
...
41
1
...
Definitions of
2
...
5
...
4
...
are functions; whereas,
is called composed with
etc
...
Warning: The above are definitions of new functions, having nothing to do with the “distributive”
property for variables
...
Examples
,
Example 1: Find the output formulas for
where
and
...
Find:
and
Solution
a)
b)
c)
2
43
-6
-8
Let and be given by the graphs below:
Example 3:
Find:
a)
b)
c)
Solution
a)
by graphs
of and
b)
graph
of
c)
d)
=
3
graph
of
-1
44
2
d)
C
...
The domains of
looking at their output formulas!
,
,
,
etc
...
Reason output formula gives the wrong domain:
Example:
If
and
, then
However,
Note that
...
is undefined:
, but and are undefined!
3
...
45
Exercises
,
1
...
Let
3
...
...
and
...
b
...
4
...
b
...
d
...
6 Domain of Combined Functions
A
...
We will
for the next two sections
...
B
...
Find
and
,
,
, and
:
...
Intersect them in AND, i
...
find where they overlap
...
In the
case, additionally throw out any ’s where
(since the denominator can’t be zero)
47
C
...
Find
and
...
Note: By the output formula, you would get the incorrect domain of
Now we’ll do it correctly
...
intersect
Ans
48
Throw away
Throw away
...
Find
and
...
Note: As we’ll see the domain is not
2
...
(root) and
(denominator)
(root) and
(denominator)
intersect
,
Since we have , here we need to also throw out where
i
...
Throw out
Ans
:
...
50
1
...
Introduction
In this section, we will discuss how to find
from output formulas
...
Motivation
Since
, the domain must consist of
1
...
also,
[
[
]
must be accepted by
; this occurs when
is defined
]
C
...
Find
...
Find
...
Intersect
Note:
with
(overlap with AND)
...
D
...
and
Solution
1
...
Now find
,
...
Intersect:
:
from the even root, we require
Thus
was never used!
52
...
Solution
1
...
Now find
We see that
, so
, so
intersect
53
...
Intersect:
1
...
Method
1
...
Draw
from the graph
...
3
...
4
...
B
...
From the graph,
...
Mark
on the -axis of the graph of :
include
Band determined by
Throw away everything outside
include
3
...
Now read off the domain:
Ans
Example 2:
Find
where
and are given by the following graphs:
55
Solution
1
...
Mark
...
“Mutilated” graph of :
4
...
Let
2
...
Let
4
...
Find
and
...
Find
and
and
...
...
...
Let and be given by the following graphs:
a
...
Find
c
...
Find
57
6
...
Find
b
...
Find
d
...
9 Inverses
A
...
2
...
Examples
Are and inverses, where
Example 1:
and
?
Solution
Check the two conditions!
Ans
1
...
YES, and are inverses
59
Are and inverses, where
Example 2:
and
?
Solution
Check the two conditions!
1
...
Both conditions are not met, so
...
Determine if and are inverses where
2
...
Determine if and are inverses where
and
and
61
and
...
...
10 One-to-One Functions
A
...
We might ask the following question:
Does there exist a function , so that and are inverses?
B
...
e
...
Let’s consider
,
for example
...
Notice what does to -3 and 3:
-3
9
3
Since sends both -3 and 3 to 9, if we had a that worked, would have to send 9 back to
-3 AND 3, but functions can’t do that!
Moral: We see that for a function to have an inverse, it can not send two or more ’s to the
same number
...
11)
...
Definition of a One-to-One Function
A function is called one-to-one if it is impossible for different inputs to get sent to the
same output
...
3
4
6
3
6
5
7
5
7
9
8
one-to-one
D
...
Definition: See if it is impossible for different ’s to go to the same
...
Graphical: Horizontal Line Test
(if any horizontal line hits the graph more than once, it is not one-to-one)
not one-to-one
3
...
Set
b
...
If
(only), then
is one-to-one; otherwise, it is not
63
E
...
Set
2
...
Set
:
LCD=
2
...
Set
2
...
2
...
...
...
...
6
...
7
...
8
...
66
1
...
Notation
1
...
2
...
B
...
Verify that is one-to-one
...
Set output
Note:
and solve, if possible, for
67
(the input)
...
Examples of the Algebraic Method
Example 1:
where
Find, if possible, the output formula for
Solution
1
...
2
...
68
Example 2:
Find, if possible, the output formula for
where
Solution
1
...
Let
and solve for :
...
D
...
Thus, if we have the graph of
, and we want to evaluate at a point, we put in the -value and take the corresponding
-value as the answer
...
An Example
Let be given by the following graph:
Evaluate:
a
...
Solution
Note: actually has an inverse, since it passes the Horizontal Line Test
a
...
: when
: when
,
, so
,
, so
70
Exercises
1
...
b
...
d
...
f
...
h
...
If is one-to-one, find the output formula for
71
, where
where
...
Let be given by the following graph:
Evaluate:
a
...
c
...
12 Inverse Functions II: Reflections
A
...
Thus, for
graphical
purposes,
we
can
get
a
“non-sideways”
version
of
the
graph
of
by
switching
the and coordinates
...
Note: When we want to consider this alternate version of the graph of , we indicate that
in our output formula as well
...
formula is
Now let’s see what this means geometrically
...
Graph of the Inverse Function
To get the graph of , we switch the coordinates of each point on the
graph of
...
73
C
...
If
is on
, then
is on
...
Connecting these two points, we cut the line
3
...
Since the slope
of the line
to the line
...
...
To show that
same distance from as the point
is
...
Distance from
to
:
7
...
74
is the
, so the graph of
is the
1
...
Domain and Range of
1
...
To find
: find
This comes from the idea that
takes -values back to -values
...
The output formula
domain and range
...
will give the wrong
Examples
Example 1:
is invertible and
...
Solution
1
...
Note: The above output formula suggests that
the right answer
...
75
, but this is not
2
...
Thus
3
...
is invertible and
...
: Find
Now
Thus
Hence
2
...
...
Let
2
...
Let
...
Find
,
,
...
...
1
...
Motivation
If is useful, but not invertible (not one-to-one), we create an auxiliary function
similar to , but is invertible
...
that is
Capital Functions
Given , not invertible
, we define
invertible
...
[
2
...
is one-to-one
b
...
produces the same outputs as , so the output formula is the same]
is called a capital function or principal function
Method for Constructing Capital Functions
Given , we try to determine what to cut out of
the same range
...
Examples
Example 1:
...
Solution
To see what is going on, let’s look at the graph of :
Note:
...
Ans
One solution is
Another solution is
Note: Each such solution can be loosely referred to as a “branch”
...
Construct a capital function
...
]
...
Ans
One solution is
Another solution is
80
E
...
By construction, the capital functions are invertible
...
Since
, it retains the useful information from the original function
...
The inverse of the capital function,
of the original function one can get
...
, serves as the best approximation to an inverse
Inverses of the Capital Functions
Here given a function , we examine
Example 1: Let
Also, give
and
...
...
Using the left branch,
;
81
and find
...
In this case,
make smaller than -3
...
...
Using the right branch, determine
Solution
Let’s look at the graph of :
82
and find
...
Using the right branch,
Note 2:
;
and
Now find
:
Then
OR
OR
OR
...
In this case,
,
so
we
need
the
equation
giving
-values
that are 2 or bigger
...
Hence,
...
Let
...
Also, determine
2
...
Let
determine
...
...
Let
Also, determine
...
85
and find
and find
and find
...
...
Also,
and find
...
1 The Reciprocal Function
Let
...
We can plot this by making a table of values
...
undefined
87
undefined
, we pick a lot
For very large numbers
(like ) and very negative numbers (like ), the graph
approaches the -axis, but does not cross
...
These “imaginary” approaching lines are called asymptotes
...
88
or axes, as in this function, we often draw
2
...
Definition of a Rational Function
is said to be a rational function if
, where and are polynomial functions
...
B
...
Asymptote Types:
1
...
horizontal
3
...
curvilinear (asymptote is a curve!)
We will now discuss how to find all of these things
...
Finding Vertical Asymptotes and Holes
Factors in the denominator cause vertical asymptotes and/or holes
...
Factor the denominator (and numerator, if possible)
...
Cancel common factors
...
Denominator factors that cancel completely give rise to holes
...
D
...
Solution
Canceling common factors:
factor cancels completely
vertical asymptote with equation
Find the vertical asymptotes/holes for where
Solution
Factor:
hole at
factor not completely canceled
Example 2:
90
...
Finding Horizontal, Oblique, Curvilinear Asymptotes
Suppose
If
1
...
degree top
3
...
Examples
Example 1:
Find the horizontal, oblique, or curvilinear asymptote for where
Solution
degree top
Ans
degree bottom
horizontal asymptote with equation
91
Since
...
Example 2:
Find the horizontal, oblique, or curvilinear asymptote for where
Solution
degree top
Since
Ans
degree bottom
...
Thus
...
, we have an oblique or curvilinear asymptote
...
Example 4:
Find the horizontal, oblique, or curvilinear asymptote for where
Solution
degree top
Since
degree bottom
...
Now long divide:
Since
, we have that
Throw away
Ans
defines a curvilinear asymptote
93
G
...
As the graph of a function approaches a vertical asymptote, it shoots up or down toward
approach
approach
2
...
approach
or
approach
approach
approach
3
...
94
...
Find the vertical asymptotes and holes for where
a
...
c
...
Find the horizontal, oblique, or curvilinear asymptote for where
a
...
c
...
f
...
3
...
4
...
What is the domain of the
function?
5
...
95
2
...
Strategy
1
...
2
...
3
...
Note: The graph may not cross vertical asymptotes, but may cross others
...
B
...
Asymptotes:
Vertical:
Horizontal:
2
...
-intercept: set
:
Now graph an initial rough sketch:
Now
need to plot enough points to see what is going on
...
Then pick a few others to
see what is going on:
We plot these points on the grid we already made
...
97
Ans
Example 2:
Graph , where
Solution
1
...
Top:
Bottom:
Rational Root Theorem:
factors of
factors of
Rational Candidates:
98
Note:
Thus
Thus,
Simplifying,
0
...
Thus, we have vertical asymptotes with equations
Also, we have a hole at
and
, we have a
2
...
...
is a factor
...
Now graph an initial rough sketch:
y
x
hole on graph here
Now plot some more points using the output formula
:
Note: In fact, the point from thetable
above is the location of a hole, since
...
Then we connect the points using
the asymptote behavior
...
Asymptotes:
We first have to factor
...
However
the zeros are complex
...
101
Now find it by algebraic long division:
...
2
...
...
Divide it out using synthetic division:
102
...
Thus we have one -intercept,
-intercept: set
:
Now graph an initial rough sketch:
103
...
Now plot some more points using the output formula
:
We plot these points on the grid we already made
...
Ans
104
Example 4:
Graph , where
Solution
1
...
...
Now find it by algebraic long division:
Hence
Thus
...
105
2
...
...
:
-intercept: set
This intercept is not a point either, since we found it by setting
is where we said we had a hole
...
Note: Since this function
is even, we need only make our table with negative
values
...
We plot these points on the grid we already made, along with the -axis reflected
points (even function)
...
Ans
hole
107
Exercises
Graph the function , where
1
...
3
...
108
109
110
Chapter 3
Elementary Trigonometry
3
...
Circles
Standard Form:
1
...
radius:
3
...
Examples
Findthe
center, radius, circumference,
...
Example
1:
and intercepts of the circle, where
Solution
center:
radius:
circumference:
-intercepts: set
:
Thus there are no -intercepts
...
Graph:
circumference:
Example
2:
Find the center, radius, circumference,
...
Solution
center:
radius:
circumference:
:
-intercepts: set
Thus the -intercepts are
113
...
Graph:
y
circumference
x
Note: This special circle is called the unit circle
...
Revolutions on the Number Line
We put a unit circle on the number line at , and allow the circle to “roll” along the number
line:
circumference
Note: The circle will touch (at the point
the right one complete revolution
...
Examples
Where does the circle touch the number line if the circle rolls
...
Find the standard form of the equation of the specified circle:
a
...
center:
c
...
Find the center, radius, circumference,
a
...
and intercepts, and sketch the circle where
b
...
Why does
not define a circle?
4
...
a
...
Left Half a Revolution?
c
...
Right
a Revolution?
e
...
2 The Wrapping Function
A
...
circumference
Let (theta) be the number line variable
...
: wrapping function
: point on the unit circle where
on the number line wraps to
circumference
Note:
is an ordered pair in the plane and not a number
...
Strategy
To easily and semiaccurately locate
, we use the following guidelines:
1
...
wraps
2
...
For instance, if (in lowest terms), we want to find , then
we divide the semicircle into equal sized wedges and count to the correct wedge,
namely the th one
...
To locate
recognize that
when is an integer, we use the number line as a guide and
...
119
C
...
Solution
We divide up the upper semicircle into four equal wedges and count over to the
third wedge:
120
Example 2:
on the unit circle
...
We divide up the lower semicircle into six equal wedges and count clockwise to the
fifth wedge:
121
Example 3:
Locate and mark
on the unit circle
...
Note that
...
Then we need to go
more
...
122
Exercises
1
...
Locate and mark on
a
...
b
...
c
...
d
...
e
...
f
...
g
...
h
...
2
...
3 The Wrapping Function At Multiples of
A
...
Evaluation
Locate
and read off the coordinates
...
Examples
Example 1:
Evaluate
Solution
Ans
125
Example 2:
Evaluate
Solution
Ans
126
Example 3:
Evaluate
Solution
Ans
127
3
...
Introduction
Evaluating
evaluating
for
when
We will derive the
being a multiple of
is a multiple of
...
However, we need a rule for
rule in six easy steps
...
Derivation of the Rule
Step 1: Note that if is a multiple of
(lowest terms), then
128
is in one of 4 spots
...
By symmetry, we get the rule
...
Rule
By symmetry, the and coordinates of
with appropriate signs
...
Strategy
Locate the point on the unit circle, and then use the rule based on the picture
...
Examples
Example 1:
Evaluate
Solution
First locate
on the unit circle:
Since we see that to locate the point, we must have positive
have that
Ans
132
and negative , we
Example 2:
Evaluate
Solution
First locate
on the unit circle:
Since we see that to locate the point, we must have negative
have that
Ans
133
and positive , we
Example 3:
Evaluate
Solution
First locate
on the unit circle:
Since we see that to locate the point, we must have negative
have that
Ans
134
and negative , we
Exercises
Evaluate the following exactly:
1
...
3
...
5
...
7
...
9
...
11
...
135
3
...
Derivation of the , Rule
Step 1: Locate
and
Step 2: Form a central triangle, and label the inside angles
, , and
Notice that since the upper
semicircle has been cut into 3 equal pieces,
we have that
...
and
Step 4: Since
, the triangle is equiangular
...
137
Step 5: By symmetry, the axis bisects the triangle into two with top edge length
Step 6: We examine triangle
We can use the Pythagorean Theorem to find :
138
Step 7: We now have the following picture, from which we can read off
Thus
...
B
...
These give the and coordinates of for being a multiple of or with
appropriate signs
...
Strategy
Locate the point on the unit circle, and then use the rule based on the picture
...
Examples
Example 1:
Evaluate
Solution
First locate
on the unit circle:
Then we draw the triangle and label it:
Since we see that to locate the point, we must have negative
have that
140
and positive , we
Ans
Example 2:
Evaluate
Solution
First locate
on the unit circle:
Then we draw the triangle and label it:
141
Since we see that to locate the point, we must have positive
have that
Ans
Example 3:
Evaluate
Solution
First locate
on the unit circle:
142
and negative , we
Then we draw the triangle and label it:
Since we see that to locate the point, we must have positive
have that
Ans
143
and positive , we
Exercises
Evaluate the following exactly:
1
...
3
...
5
...
7
...
9
...
11
...
144
3
...
Definitions
Let
...
and then use the definitions
Note: For some values of , we may get division by zero upon evaluating a trigonometric
function
...
We will discuss this issue later in the next
section
...
Examples
Example 1:
Find
Solution
:
We first find
...
, so
147
Example 3:
Find
Solution
:
We first find
Thus
Now
Ans
...
, so
149
Example 5:
Find
Solution
:
We first find
Thus
Now
Ans
...
, so
...
151
Exercises
1
...
b
...
j
...
l
...
e
...
g
...
c
...
o
...
m
...
Suppose
and
...
7 Domain and Range of the Trigonometric Functions
A
...
Domain:
Since is defined for any
restrictions
...
2
...
Hence we can see that
, namely -1; the
...
-coordinate
B
...
Domain:
, we have
Given
does this happen?
...
When
happens here
Thus
is undefined for
What is this in interval notation? To see it, let’s plot the allowed values on a number
line:
Thus
:
-
Note: Each interval has an endpoint being an “odd multiple of ”
...
2
...
C
...
Domain:
, we have
This is similar to tangent
...
When does this happen?
Thus
is undefined for
happens here
155
...
2
...
D
...
Domain:
Given
, we have
does this happen?
...
When
happens here
So similar to tangent,
...
Range:
On the right semicircle,
On the left semicircle,
ranges from down to , so
ranges from near to
156
, so
ranges from up to
ranges from
up to
...
Hence
...
Cosecant
1
...
Now
...
2
...
When
F
...
and are undefined at odd multiples of
...
and
are undefined at multiples of
...
8 Trigonometric Functions: Periodicity
A
...
B
...
The smallest
such value of that makes the function periodic is called the period
...
Periodicity of the Wrapping Function
, so the wrapping function is periodic
...
Section 3
...
Periodicity of the Trigonometric Functions
Since the trigonometric functions are defined in terms of , they are also periodic, and
repeat every
...
so, in particular, tangent and cotangent
E
...
9 Trigonometric Functions: Even/Odd Behavior
A
...
161
B
...
Even Functions:
even
even
2
...
Examples
Example 1:
Suppose
...
Solution
Ans
-
162
Example 2:
Suppose
...
Exercises
1
...
Use even/odd relationships to simplify
...
Use even/odd relationships to simplify
...
Find
and odd
...
Suppose
3
...
10 Elementary Trigonometric Relationships
A
...
B
...
2
...
and
and
and
165
C
...
2
...
The Pythagorean Identity
Note:
(because we have a unit circle)
Since we have that
Warning:
and
Shorthand:
, the
equation becomes
does not mean
;
166
means
E
...
Examples
Example 1:
If
, what are the possible values of
Solution
We use Pythagorean I:
Thus,
Ans
168
?
Example 2:
If
, what are the possible values of
Solution
We use Pythagorean II:
Thus,
Ans
169
?
Exercises
1
...
Know
3
...
If
5
...
If
7
...
If
...
Find
...
...
, what are the possible values of
, what are the possible values of
?
?
, what are the possible values of
?
, what are the possible values of
, what are the possible values of
?
170
?
171
172
Chapter 4
Graphing Trigonometric Functions
4
...
Graph of
Since
we are now familiar with the function, we may write
the and in the equation with the and coordinates in the output to
and not confuse
...
Since we know that is -periodic,
we only need to make a table from to
...
B
...
Sinusoidal Graphs
Oscillatory graphs like
take the form:
and
are called sinusoidal graphs
...
Features
1
...
graph bounces between
and
instead of and (stretching)
b
...
Period:
a
...
accounts for horizontal stretching
3
...
Examples
Example 1:
Let
...
Solution
Amplitude:
Period:
Example 2:
Let
...
Solution
Amplitude:
Period:
Phase Shift:
176
In the next section, we will look at how to graph sinusoids using a “modified” HSRV
strategy
...
Sketch the graph of
by plotting points between
and
2
...
b
...
3
...
b
...
d
...
178
...
2 Graphing Sinusoids
A
...
Set
2
...
This gives the start of one cycle
...
This gives the end of one cycle
...
Draw one cycle with amplitude
...
If
is negative, flip across the -axis
...
To get the final graph, perform the vertical shift using the parameter
...
Examples
Example 1:
Graph , where
...
2
...
4
...
5
...
Ans
180
Example 2:
Graph , where
Solution
1
...
3
...
4
...
Shift up 3 to get final answer
181
, is nega-
Ans
centerline
Example 3:
Graph , where
Solution
1
...
182
3
...
Reflect across the -axis:
183
5
...
Graph , where
2
...
b
...
d
...
f
...
h
...
Find
...
3 Sinusoidal Phenomena
A
...
Examples: tides, yearly precipitation, yearly temperature
B
...
Find
(amplitude): Let
2
...
Find :
a
...
then let
(since
time from max to min
)
4
...
first find the phase shift:
b
...
Example
(Tides)
The depth of water at the end of a dock varies with the tides
...
5
4
...
5
2
...
4
am pm
2
...
5
Solution
We model using
1
...
Find :
max
min
3
...
period,
b
...
Find :
a
...
Ans
, where
187
is time in hours past midnight
...
The water at the end of dock varies with the tides
...
Using the data below, construct a sinusoidal model for the water depth,
, in terms of the number of hours, , past midnight
...
5 m
4
...
4 m 4
...
5 m
am
am pm
2
...
5 m
2
...
Measurements of the water depth were taken
every 2 hours and recorded
...
Time
am
am
am
am
Depth
3
...
4 m 3
...
6 m 2
...
6 m
3
...
The table below gives the recorded high temperature as measured on the th of the indicated
month
...
Month
Temperature
JAN
FEB
F
F
MAR APR MAY
F
Month
JUL
AUG
SEP
Temperature
F
F
F
JUN
F
OCT
NOV DEC
F
188
F
F
F
F
4
...
Graph of
Since
, whenever
we get vertical asymptotes:
Thus the graph of
happens here
has vertical asymptotes at odd multiples of
...
undefined
189
to
,
B
...
happens here
This occurs at multiples of , and making a similar table of values, the graph looks like:
190
C
...
Also, since
Furthermore, since
functions
...
Graph
, as in
and
or
or
, we get vertical asymptotes at multiples of
...
2
...
Doing this for
and
yields:
191
192
4
...
General Tangent
Here
we
use
the
idea
that
has
vertical
asymptotes
at
and
-intercept halfway in between, and we use the fact that is -periodic
...
Set
and
to find the location of two vertical asymptotes
...
Put an -intercept halfway in between the two asymptotes
...
Draw in a “copy of
4
...
, flip about -axis
...
Shift centerline up
units
...
General Cotangent
This
except the asymptotes
strategy as general tangent,
forcotangent
are at
has the same
and
, so for
and
in the
, we set
above strategy
...
Examples
Graph , where
Example 1:
Solution
1
...
-intercept (one of): halfway in between
and
3
...
No reflections
5
...
2
...
Now draw in the “copy of
” and make periodic:
4
...
Move Up 2
...
2
...
Now draw in the “copy of
” and make periodic:
4
...
5
...
Comments
1
...
2
...
6 Graphing General Secant and Cosecant
A
...
Graph
2
...
3
...
B
...
,
we use the same strategy as above, except we first graph
C
...
First graph
a
...
:
199
2
...
3
...
First graph
a
...
201
centerline
2
...
Move Down 1
202
Ans
centerline
D
...
For general secant/cosecant, amplitude is undefined
...
Period:
203
Exercises
1
...
b
...
d
...
Graph , where
a
...
c
...
f
...
h
...
204
4
...
Introduction
Sometimes a trigonometric function gets multiplied by another function called a damping
factor
i
...
is the damping factor
here
Damped trigonometric functions involving sine and cosine are straightforward to graph
...
Considering
all possible in the same way, we see that the function oscillates between
(the damping factor)
...
e
...
However, in this section, we consider all viable
factors
...
Graphing Strategy
1
...
2
...
205
and
C
...
b
...
206
Now draw in the damping curves
and
, then modify:
Ans
Example 2:
Graph , where
Solution
First graph
a
...
:
207
c
...
2
...
4
...
209
4
...
Simple Harmonic Motion
An object that oscillates in time uniformly is said to undergo simple harmonic motion
...
Frequency
1
...
Frequency, :
“oscillation speed” (how many cycles per time)
Units: inverse units of time, typically s , also called Hertz (Hz)
210
3
...
Examples
Example 1: An object in simple harmonic motion is described by
...
Time is measured
in seconds and displacement is measured in meters
...
Example 2:
Find a model for simple harmonic motion satisfying the conditions:
Period: s
Maximum Displacement: m
Displacement at
: m
Solution
Since the object starts at maximum displacement, we use the cosine model:
Now
Ans
, and
...
Consider an object in simple harmonic motion
...
Find the period, frequency, angular frequency, and maximum displacement, when the
motion is described by:
a
...
c
...
2
...
b
...
d
...
Describe
the motion of a spring whose
displacement from equilibrium is described
physically
by
...
What physical condition might give rise to such a damping
factor?
214
215
216
Chapter 5
Trigonometric Identities
5
...
Review
1
...
b
...
and
and
and
2
...
b
...
Pythagorean Identities
Pythagorean I:
Pythagorean II:
Pythagorean III:
4
...
Even Functions:
even
even
b
...
Simplifying/Factoring
1
...
2
...
3
...
Common Theme: Look for opportunities to use the Pythagorean Identities by looking for
squared trigonometric functions
...
Examples
Example 1:
Simplify
Solution
Use Pythagorean I:
Thus
to replace
reciprocal
Ans
219
with
...
Simplify
2
...
Factor and simplify
4
...
Factor
8
...
Simplify
10
...
Simplify
12
...
Simplify
14
...
Factor
6
...
Factor and simplify
16
...
222
...
2 Verifying Trigonometric Identities
A
...
No matter what the input is,
the equation works (provided the expressions are defined)
...
Conditional equations are equations that only work for a few values of
(input)
Examples:
(only the solution
(only the solutions
works)
and
work)
We solve (conditional) equations, but we verify identities
...
However, when we verify identities, we have the answer already, in some sense
...
The method to do so is radically
different than the method for solving equations
...
Verifying Identities
Unlike solving equations, we are not allowed to work with both sides of an identity at the
same time to verify it
...
Example: “Prove/verify” that
If we square each side, we get
we’ve done something wrong
...
C
...
Pick one side of the equation (usually the more complicated side), and ignore the other
side
...
Manipulate it, by itself, using valid laws for expressions
...
is not allowed (you don’t have an equation to balance out
the operation!)
3
...
Hence, unlike equations (conditional) where the goal is to solve to get an “answer”, you
actually know the answer to an identity already! It is the other side of the equation! Here
you know the beginning and the end, and the goal is to fill in the middle–to show how to
get from the beginning to the end
...
Verifying Trigonometric Identities
1
...
Try to implement any trigonometric identities you can think of
...
2
...
3
...
4
...
Even if you can’t see immediately what to do, try something! Dead ends sometimes give
you ideas that help you see the correct approach
...
You are
trying to fill in the middle
...
Examples
Example 1:
Verify the identity:
Solution
Start with the left side:
(Now use Pythagorean I:
225
)
Thus we reached the right side, so we are done
...
226
Example 3:
Verify the identity:
Solution
Start with the left side:
(adding)
(multiply out bottom)
(use Pythagorean I)
(use reciprocal identity)
Thus we reached the right side, so we are done
...
Note: Sometimes multiplying top and bottom by something that causes a Pythagorean
identity is a good plan, as in the next example
...
229
)
Example 6:
Verify the identity:
Solution
Start with the left side:
(LCD)
(convert to sines and cosines)
(use Pythagorean I)
Thus we reached the right side, so we are done
...
2
...
4
...
10
...
15
...
12
...
11
...
5
...
231
16
...
18
...
20
...
3 Sum and Difference Formulas I
A
...
Thus as calculated in Step 2 is
the same as the distance between and
...
234
, and
Step 5: Set the two expressions for equal, and use algebra
Hence, we have that
B
...
”
C
...
Justification:
2
...
Derivation of
Using the cofunction identity for
Thus
E
...
Formula for
Writing
as
, and then expanding and simplifying (Exercise), we get
Comments:
1
...
If they are not defined, then you need to simplify the expression the long way,
using
238
G
...
2
...
4
...
Co-sine is short for complementary sine, that is cosine is -complementary to sine
...
239
H
...
Comments
1
...
2
...
Find
2
...
Find
4
...
Find
6
...
Find
8
...
Find
10
...
Find
12
...
Simplify
to obtain the formula
14
...
243
5
...
Summary
1
...
only works when
2
...
244
B
...
Simplify
2
...
Simplify
4
...
Simplify
6
...
Show that
and
8
...
b
...
d
...
f
...
5
...
Derivations
1
...
3
...
Alternate Cosine Form: Replacing
a
...
248
is defined;
B
...
The above formulas are called the double angle formulas
...
C
...
Simplify
2
...
Derive a “triple-angle” formula for
4
...
Simplify
251
5
...
Derivation for
1
...
, and
, so
Then
,
3
...
Summary
, so
252
C
...
with
253
)
Example 2:
Graph , where
Solution
Use the power reducing formula for sine!
Thus, we graph where
1
...
:
3
...
Reflect about the -axis:
5
...
Rewrite
2
...
Rewrite
4
...
without powers
...
5
...
Graph , where
without powers
...
7 Half-Angle Relationships and Formulas
A
...
The choice of root will depend on the
specific value of that is used in a problem
...
Half-Angle Formulas for tangent
Here we will be able to get an explicit formula for with no sign ambiguity
...
for sine)
Now we will obtain another formulation:
(Pythagorean I)
(diff
...
However,
provided these values are not used, the
second expression is often simpler to use
...
Evaluation Examples
Example 1:
Find
Solution
First locate
:
258
work
Use the half-angle relationships for sine and cosine:
Also
Ans
Example 2:
Then,
, and since
Find
, we have
Solution
Here we use the half-angle formula for tangent:
259
, we have
...
(We could have used the other half-angle formula here, and it would result in the
same answer, but it would involve slightly more work)
Ans
260
Exercises
1
...
Find
3
...
Find
5
...
Find
261
5
...
of squares)
Exercises
Verify the following trigonometric identities:
1
...
5
...
3
...
265
5
...
Derivations
Consider the sum and difference formulas for sine:
Adding these two equations, we get:
Dividing by 2, we get:
By similar methods, we also get:
B
...
C
...
Solution
Use
Then
Ans
268
Exercises
1
...
Evaluate
3
...
Evaluate
5
...
6
...
Express
as a sum or difference
...
8
...
5
...
Derivations
Consider the sum and difference formulas for sine:
Adding these two equations, we get:
Let
Since
and
Then
Thus
and
and
...
...
The derivations of the other sum to product formulas are similar
...
Summary
Note: Again, it is more useful to remember how to get the sum-to-product formulas than it
is to memorize them
...
Examples
Express
Example 1:
as a product
Solution
Use
Thus
Ans
:
(even identity for cosine)
271
Example 2:
Express
as a product
Solution
Use
Ans
:
Thus
272
Exercises
1
...
Express
as a product
...
Express
4
...
as a product
...
5
...
5
...
2
...
4
...
1 Capital Trigonometric Functions
A
...
The -plane is divided into 4 quadrants by the and axes
...
The Six Trigonometric Functions
To motivate what comes next, let us first review the graphs of the six trigonometric functions
...
Motivation
All six trigonometric functions fail the horizontal line test, so are not one-to-one/invertible
...
D
...
This is
not the only choice, but it is the most obvious choice
...
We call it
Thus
...
E
...
1
...
3
...
5
...
F
...
The only difference between the behavior of the capital trigonometric functions and the
ordinary trigonometric functions is the restricted domain
...
Like all capital functions, the capital trigonometric functions are invertible
...
2 Capital Trigonometric Problems I
A
...
Rewrite the capital trigonometric functions as the ordinary trigonometric functions with
the appropriate domain restriction
...
Use ordinary trigonometric identities to solve the problem
...
Use the restricted domain to remove the ambiguity in sign
...
Examples
Example 1:
You know
...
Solution
1
...
To get
from , we use
285
:
3
...
Find
...
, so
286
2
...
Use the restricted domain to try to remove the sign ambiguity:
Since
, we are in the region marked:
Here
Ans
, so
287
Exercises
1
...
You know
3
...
You know
5
...
Find
...
...
Find
...
...
Find
6
...
...
3 Capital Trigonometric Problems II
We consider some more complicated examples
...
Find
Solution
1
...
We know that
To get
, we use
, so we need
289
...
Use the restricted domain to try to remove the sign ambiguity:
Since
, we are in the region marked:
Here
, so
...
Thus,
Ans
290
Example 2:
Know
Find
...
2
...
291
3
...
However, our original problem was to find
...
Find
Solution
1
...
To get , we use
3
...
However, we were originally given
, so in particular
...
Thus
Ans
and so we have that
...
Find
Solution
1
...
To get
, there are many different methods that can be used
...
One method:
First find
...
Another method:
Use
to find
Here let us arbitrarily use the first method
...
...
Then
3
...
Thus we have no initial help!
However, since we were originally given
Thus
...
This can not happen in quadrant I, so we must be in quadrant IV
...
Exercises
1
...
Know
3
...
Know
5
...
Know
9
...
Know
...
6
...
Know
...
Know
12
...
Find
...
Find
...
Find
...
...
Find
...
Find
...
...
...
...
and
...
6
...
Introduction
Even though the ordinary trigonometric functions are not invertible, the capital trigonometric functions are (by design)
...
B
...
Graphs of the Inverse Trigonometric Functions
299
300
D
...
Warnings:
a
...
b
...
Be careful of this in problems
...
Some authors are lazy and write
To avoid confusion, write
if that is what is intended
...
, ,
,
, ,
are sometimes written
2
...
In that context, inverse sine,
pronounced “arc-sine” when it is written as
...
Evaluation
We can evaluate inverse trigonometric functions if the output is a multiple
of , , , , or
...
Remember the range of the inverse trigonometric function!
301
F
...
coord
coord
, so we have coord
and coord
Thus
Ans
...
Example 4:
Evaluate
Solution
Note:
and are not inverses
...
Now
Thus we ask “which
Ans
has
with
as the -coordinate?”
Example 5:
Evaluate
Solution
Note:
Now
and
are not inverses
...
Thus we ask “which
Ans
has
304
with
as the -coordinate?”
Example 6:
Evaluate
Solution
Here we can’t evaluate
Thus, in this case,
Thus we have
directly, but we notice that
...
, since
and
are inverses!
Note: We couldn’t do this in the previous examples, since the numbers weren’t in
the domain of the capital function
...
2
...
4
...
6
...
8
...
10
...
12
...
5 Inverse Trigonometric Problems
A
...
Define the inverse trigonometric function output to be
...
Rewrite the definition with no inverse trigonometric function by applying the appropriate capital trigonometric function to each side
...
Recast the problem as a capital trigonometric function problem, and solve it
...
Examples
Example 1:
Find
Solution
1
...
Then
...
3
...
...
307
b
...
Since
,
Here
Ans
, so
308
Example 2:
Find
...
Let
2
...
3
...
...
Find
...
Since
,
Here
, so
Then,
Ans
310
Example 3:
Find
Solution
1
...
Then
...
3
...
b
...
However
Thus
...
...
...
Ans
Example 4:
Find
Solution
1
...
Then
...
3
...
...
312
b
...
Since
, so we need
,
Thus
, so
However we want , so
Ans
313
...
...
Let
2
...
...
Thus we have the capital trigonometric problem:
You know
a
...
)Find
:
...
Find
and
Thus we need
b
...
and
...
)Find
, so
:
315
...
Exercises
1
...
Find
3
...
Find
5
...
Find
7
...
Find
13
...
Find
15
...
Find
12
...
Find
9
...
Find
17
...
Find
318
6
...
Single Function Method
This method is used when each side only contains one inverse trigonometric function
...
Let
one side
...
Manipulate this equation to get rid of the inverse trigonometric function and reduce the
resulting capital trigonometric function to an ordinary trigonometric function with domain
restriction
...
Use regular trigonometric identities to simplify the resulting equation
...
Reverse the process to get
other side
...
Examples
Example 1:
Verify the identity:
Solution
1
...
Then
...
[odd identity for sine]
319
4
...
Let
2
...
Since
4
...
, so
is impossible anyway, so
320
C
...
Assuming that the sum is on the left hand side of the identity
...
Simplify sum as an inverse trigonometric problem to get an identity for sum
...
Use the domain restriction to get an appropriate identity for the original sum
...
instead of
...
Examples
Example 1:
Verify the identity:
Solution
1
...
...
Find
...
Find
:
Now use the domain restriction to eliminate sign ambiguity:
Since
,
Thus
323
, so
...
Now we need to use the domain restriction to get the original identity:
Now
and
Thus we have that
Since
only value of
Aside:
which is easier, as follows:
Let
Then
, so
...
(cofunction identity)
324
Then solving for , we have
Hence we verified that
...
Example 2:
Verify the identity:
Solution
Since is more natural here
...
Simplify
Let
Then
:
and
and
...
Hence we have the following capital trigonometric problem to solve:
Know
and
325
...
Now
and
Also
Hence we have the identity:
2
...
...
It only remains to determine which of the two identities is correct
...
Since
with
we must be in quadrant I
...
since
Exercises
Verify the following inverse trigonometric identities:
1
...
3
...
7
...
9
...
5
...
[Hint: Start with the right hand side]
328
6
...
Summary
1
...
b
...
Cofunction Inverse Identities
a
...
c
...
Reflection Identities
a
...
c
...
e
...
B
...
In particular,
1
...
3
...
In fact, using the identity,
inverse sine button only!
330
, we can reduce the need to that of an
Then
1
...
3
...
5
...
8 Solving Trigonometric Equations I
A
...
functions, i
...
332
B
...
Since
, we have that coord
...
However, more than that, by adding we get two more solutions
...
Strategy
1
...
2
...
3
...
Note: In situations where more than one type of trigonometric function occurs in an equations, we try to either
a
...
get rid of one of the trigonometric functions via trigonometric identities
...
Examples
Example 1:
Solve
for
Solution
coord
334
Ans
Solve
Example 2:
for
Solution
Let
, to make a standard quadratic equation
...
By the Zero Product Principle:
coord
or
or
or
coord
Ans
336
Example 3:
Solve
for
Solution
Use Pythagorean I to eliminate the trigonometric function
By the Zero Product Principle:
coord
or
or
or
or
coord
It is impossible for the -coordinate to be , so coord
337
...
Comments
1
...
For example, if we consider the answer to Example 1:
it can be written as
338
since
since
and
differ by
and
and
differ by
...
2
...
See the examples that follow
...
More Examples
Example 1:
Solve
for
Solution
(Diff
...
Consider
:
Thus
Hence,
II
...
Consider
:
Thus
Putting all of these solutions together, we get
Thus, we have (upon reducing),
Ans
, we don’t divide by 5 until the very end
Note: For
after we have all the solutions via adding
...
This motivates the following trick:
1
...
2
...
Now do it:
1
...
2
...
4
...
6
...
8
...
11
...
13
...
344
14
...
16
...
345
6
...
to solve
trigonometric equations
...
However, we can use the double angle formula
...
348
Thus,
Now
we need to do the check
...
Check:
?
Thus substituting into the original equation:
349
?
?
?
Thus substituting into the original equation:
?
?
?
Thus substituting into the original equation:
?
350
?
X
?
Thus substituting into the original equation:
?
?
are
Thus the only initial solutions that work in the interval
Ans
Example 4:
Solve
and
for
Solution
Use the sum-to product formula:
351
is even
X
By the Zero-Product Principle:
or
Consider first
:
Here we have
Thus
352
Now consider
:
Here we have
Thus
Putting all solutions together, we get
Ans
353
Exercises
Solve the following equations for :
1
...
3
...
7
...
9
...
4
...
10 Harmonic Combination
A
...
We will “reduce” this as follows:
1
...
Since
Thus
,
:
is on the unit circle
...
Thus
355
4
...
B
...
C
...
, where
In fact, we can simplify
:
Thus
...
2
...
Ans
Solve
Example 3:
for
Solution
Compress the harmonic combination
...
...
Then we have
Ans
359
:
Exercises
1
...
b
...
d
...
Graph , where
a
...
3
...
b
...
1 General Angles
A
...
363
We have solutions at
:
and
:
:?
We need to find
:
chords are the same, so
the corresponding arcs
are congruent
364
length
of arc is
Thus
:
...
However, let us reconsider the original picture:
Note: If we consider the two triangles, we know that the legs of the two triangle are congruent, since both have length and the hypotenuse of the two triangles are congruent, since
both have length (unit circle)
...
Thus the inner angles of the triangles are the same
...
Goal: Connect arc length to angles
...
Radian Measure of Angles
On the unit circle, we define the radian measure of an angle to be the signed arc length on
the circle as a number (in the same fashion as the wrapping function)
...
Comments
1
...
Special Angle: radians
D
...
Hence, we have the arc length formula:
E
...
F
...
...
To convert degrees to radians: multiply by
...
To convert radians to degrees: multiply by
...
Radian measure for an angle is just a number!
G
...
b
...
Solution
b
...
c
...
b
...
b
...
Comments on Terminology
1
...
Coterminal angles differ by a multiple of (or by )
370
2
...
Complementary Angles: Angles that differ by
I
...
If an angle is
given in degrees, we need to convert to radians first before using the arc length formula
...
m
Solution
We first need to convert to radians:
Now use the arc length formula:
Since
Ans
, we have that
m
m
372
m
Example 2:
m
Find the central angle of a circle of radius
m
m
Solution
Ans
(in radians)
373
m that cuts off an arc length of
Exercises
1
...
b
...
d
...
f
...
Convert the following to degrees:
a
...
c
...
3
...
4
...
5
...
6
...
7
...
A
...
and
Then we have that
...
B
...
Examples
Example 1:
Given the right triangle:
Find ,
,
Solution
We first get the hypotenuse via the Pythagorean Theorem:
Now use the right triangle definitions:
opp
...
adj
...
opp
...
377
Example 2:
Given the right triangle:
Find ,
,
Solution
We first get the third side via the Pythagorean Theorem:
Now use the right triangle definitions:
adj
...
opp
...
hyp
...
378
D
...
e
...
We can solve a right triangle if we have at least the following information:
1
...
One side and one acute angle
...
Tools For Solving Right Triangles
1
...
2
...
Thus, if the two acute angles are and , we have that
or
...
Everything else can be found by using the trigonometric ratios
...
Typically
degree measure is used in triangle problems
...
379
F
...
We label the angles opposite the sides by ,
Note:
,
...
G
...
Some Terminology
1
...
Angle of Depression
Observer
Horizontal
Object
383
3
...
Find ,
, and
for the given right triangle:
a
...
b
...
2
...
Use a calculator and round to two decimal places
...
b
...
d
...
f
...
Consider a cube with edge length
...
Consider a cube with edge length
...
3 Lines and Angles
A
...
Now consider in relation to the slope of the line:
We have that
387
B
...
Picture:
2
...
...
To ensure that the smaller angle is chosen, no matter which line is “labeled” line 1,
388
Since
, we may write
3
...
Example:
Find the smallest
the
angle between
two lines
given by and
...
2
...
390
7
...
These are called oblique triangles
...
Law of Sines
1
...
Derivation:
Here we assume that we have an acute triangle, i
...
all angles in the triangle are
acute
...
e
...
Thus
, so
391
...
B
...
Law:
2
...
We will assume that the triangle is acute
...
We place the triangle on the , coordinate system as follows:
392
Now by the Pythagorean Theorem, we have that
(by above two equations)
(Pythagorean I)
393
C
...
Formulas:
2
...
The other can be derived similarly
...
Law of Tangents
1
...
Derivation:
(Mollweide’s Formulas)
396
(cofunction identities)
(quotient identities)
(reciprocal identity)
397
7
...
Introduction
We can solve oblique triangles using the Law of Sines and Law of Cosines if one side is
known, along with two other parts (sides or angles)
...
Cases
1
...
AAS : Two angles and a nonincluded side:
know
know
b
...
Two sides:
a
...
SAS : Two sides and an included angle:
know
know
know
3
...
Comments
1
...
2
...
SSA is called the ambiguous case
a
...
no triangle
II
...
two triangles
b
...
no triangle
II
...
, but
two triangles
IV
...
conditions for
obtuse:
I
...
one triangle
402
7
...
Strategy
1
...
2
...
If not sufficient, use Law of Cosines
...
Check your answers in one of Mollweide’s Formulas (it doesn’t matter which one)
...
B
...
If possible, try to find the largest angle first
...
This will tell you automatically that the other two angles are acute, and can help to
eliminate fake solutions
...
Remember that all three angles of a triangle add to
403
...
Examples
Example 1:
Solve the triangle:
,
,
Solution
Draw a Picture:
First find
:
Now find
:
Law of Sines:
Thus,
Then
...
Now we need to check the answer using one of Mollweide’s Formulas
...
)
Thus
Ans
,
,
405
Example 2:
Solve the triangle:
,
,
Solution
Draw a Picture:
First find
:
Law of Sines:
Thus,
Then
II
Now at
I,
so
I
, so we have
406
,
Now at II, the solution (as in the beginning
7
...
e
...
...
Case I:
Then find
:
Then find :
Law of Sines:
Case II:
Thus,
Then
...
Now we need to check the answers using one of Mollweide’s Formulas
...
)
Thus this triangle is valid
...
)
Thus this triangle is valid
...
Law of Cosines: Find
(largest angle)
Since
, we have
, so
Note: Since we found the largest angle, we know that
the other two angles are acute!
Find :
Now we can use the Law of Sines:
Thus
...
...
interval
in the
...
Use
:
Check:
?
?
Thus
,
?
(approx
...
Find c:
Law of Cosines:
Find B:
Law of Sines:
Then
...
b
...
Use
:
Check:
1st triangle:
?
?
? (approx
...
412
2nd triangle:
?
?
?
(approx
...
Ans
, ,
413
Exercises
Solve the following triangles
...
1
...
,
3
...
5
...
7
...
9
...
13
...
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
11
...
,
,
,
,
,
,
,
,
414
7
...
Oblique Triangle Formula
The area of the triangle,
, is given by
Using the same idea for other triangle heights, we have
The easy way to remember this is to take “one half the product of two sides and sine of the
included angle”
...
Heron’s Formula
Heron’s Formula for the area of a triangle is a formula that only involves the lengths of the
three sides of the triangle
...
Derivation of Heron’s Formula
(Mollweide’s Formula)
(cofunction identity)
(definition of )
416
(sum/diff formulas)
Hence
Then
Hence
(cofunction identity)
(sum formula for cosine)
...
, so
...
Examples
Example 1:
Find the area of the triangle
Solution
Use the oblique triangle formula!
Ans
(double angle identities)
Thus
Hence
418
,
,
Example 2:
Find the area of the triangle
,
,
Solution
Use Heron’s Formula:
Now
Then
Ans
, where
419
Exercises
Find the area of the triangle:
1
...
3
...
,
,
,
,
,
,
,
,
420
Selected Answers to the Exercises
0
...
3
...
1
...
no
1b
...
yes, explicit
1d
...
yes, explicit
1f
...
yes, implicit
2a
...
2f
...
4
...
1
...
1b
...
2b
...
2f
...
3
1a
...
1e
...
;
423
1
...
neither
1c
...
even
2
...
;
o
1
...
After you fully simplify:
;
;
3b
...
4b
...
8
2
...
5a
...
1
...
no
3
...
10
2
...
yes
1
...
1d
...
3b
...
13
1
...
14
2
...
;
;
2
...
vertical asymptote:
1c
...
horizontal asymptote:
2d
...
horizontal asymptote:
;
2
...
426
4
...
1
1a
...
2b
...
4d
...
2
a), c), e), g),
i), k), m), o)
,
3
...
3
...
7
...
11
...
5
1
...
6
...
9
...
1a
...
1e
...
1i
...
3
...
undefined
1o
...
9
1
...
10
1
...
5
...
4
...
period
;
3b
...
amplitude
3e
...
2
1a
...
1e
...
centerline
4
...
3
...
For instance, when
above, we get
(rather than )
...
4
...
1c
...
2b
...
centerline
2g
...
7
1
...
435
4
...
Period: s
Angular Frequency:
s
Frequency: Hz
Maximum Displacement:
1d
...
2c
...
1
1
...
3
...
5
...
7
...
9
...
11
...
13
...
5
...
3
...
6
...
9
...
5
...
3
...
2
...
5
437
3
...
2
...
6
5
...
7
1
...
5
...
5
...
5
...
5
...
4
...
6
...
4
...
3
1
...
5
...
9
...
439
12
...
4
1
...
4
...
8
...
6
...
4
...
7
...
9
...
14
...
440
16
...
6
...
3
...
8
...
13
...
16
...
9
1
...
5
...
8
...
10
1a
...
, where
, where
2b
...
442
7
...
1c
...
2a
...
3
...
6
...
2
1a
...
2a
...
2e
...
;
;
;
;
,
,
,
,
,
,
7
...
443
7
...
3
...
,
,
OR
,
,
,
7
...
,
13
...
7
1
...
444
Index
Symbols
, 210
, 20
,
42
, rule, 139
rule, 131
, 284
, 284
, 284
, 78
, 284
, 284
, 284
, 145, 376
, 145, 376
, 145, 376
, 47
, 47
, 47
, 51, 54
, 51
, 81
, 75
, 23
A
AAS Triangle, 398
acute angles, 379
acute triangle, 391
adjacent side, 376
ambiguous case, 400
amplitude, 175
angle between two lines, 389
angle of depression, 383
angle of elevation, 383
angular frequency, 210
arc length formula, 368
area, 415
ASA triangle, 398
asymptotes, 88
, 47
, 18
, 18
, 42
, 42
, 42
, 42
, 38
even
, 38
odd
, 75
, 23
, 145, 376
, 145, 376
, 145, 376
, 210
, 210
Hz, 210
, 118
, 298
, 298
, 298
, 81
, 81
, 298
, 298
, 298
, 67
, 73
, 67
, 81
445
domain of composition, 51, 54
domain of inverse trigonometric functions, 298
domain of trigonometric functions, 153
double angle formulas, 249
B
bands, 54
bearings, 384
branch, 79, 81
C
capital function, 78
capital trigonometric functions, 284
capital trigonometric problems, 285
center, 111
centerline, 182, 186
circles, 111
circumference, 111
cofunction identities, 236
cofunction inverse identities, 329
combinations, 42
combining functions, 42
complementary angles, 371
conditional equations, 223
converting degrees to radians, 369
converting radians to degrees, 369
cosecant, 145, 376
cosine, 145, 376
cotangent, 145, 376
coterminal angles, 370
crossing asymptotes, 94
cursive, 18
curvilinear asymptotes, 89, 91
E
equilibrium, 210
even and odd trig functions, 162
even functions, 35
even/odd decomposition test, 40
even/odd tests, 36
explicit functions, 15
F
factoring formulas, 9
factoring trigonometric expressions, 219
formal method, 63
frequency, 210
function composition, 42
function definition, 15
function notation, 18
function operator, 18
functions, 15
G
general cosecant, 199
general cotangent, 193
general secant, 199
general tangent,
193
graph of , 175, 280
graph of , 191, 281
graph of , 192, 282
graph of , 192, 282
graph of , 174, 280
graph of , 190, 281
graph of , 299
graph of , 300
graph of , 300
graph of , 300
graph of , 299
graph of
, 299
graphing rational functions, 96
graphing sinusoids, 179
D
damped trigonometric functions, 205
damping factor, 205
decomposition into even and odd parts,
38
degree measure, 368
degrees, 368
difference of cubes, 9
difference of squares, 9
difference quotient, 20
displacement, 210
domain, 23
domain finding, 23
domain of combined functions, 47
446
graphs of the inverse trigonometric functions, 299
graphs of the trigonometric functions,
174, 175, 190–192
H
half-angle formulas for tangent, 258
half-angle relationships for sine/cosine,
257
harmonic combination, 356
Heron’s formula, 415
Hertz, 210
holes, 89, 90
horizontal asymptotes, 89, 91
horizontal stretching, 176
horizontal translation, 29
HSRV transformations, 29
hypotenuse, 376
obtuse triangle, 391
odd functions, 35
one-to-one function, 63
one-to-one tests, 63
opposite side, 376
P
perfect square, 9
period, 159, 176, 210
periodic, 159
periodicity, 159
phase shift, 176
power reducing formulas, 252
principal function, 78
product to sum formulas, 266
Pythagorean I, 167
Pythagorean Identities, 167
Pythagorean II, 167
Pythagorean III, 167
I
identities, 223
implicit functions, 15
inverse function evaluation, 70
inverse function finding, 67
inverse functions, 67
inverse trigonometric functions, 298
inverse trigonometric problems, 307
inverses, 59
Q
quadrants in the -plane, 279
quotient identities, 166
R
radian measure, 366
radians, 366
radius, 111
range, 23
range finding, 25
range of inverse trigonometric functions, 298
range of trigonometric functions, 153
rational function, 89
reciprocal function, 87
reciprocal identities, 165
reciprocal inverse identities, 329
reciprocate, 199
reflecting across line
, 73
reflection identities, 330
reflections, 30
revolution, 115
right triangle trigonometry, 376
K
key points, 30
L
law of cosines, 392
law of sines, 391
law of tangents, 396
M
Mollweide’s Formulas, 394
mutilated graph, 54
O
oblique asymptotes, 89, 91
oblique triangle formula, 415
oblique triangles, 391
447
vertex formula, 27
vertical asymptotes, 89, 90
vertical line test, 17
vertical translation, 30
S
SAS triangle, 399
secant, 145, 376
shrinking, 29
simple harmonic motion, 210
simplifying trigonometric expressions,
219
sine, 145, 376
sinusoidal graphs, 175
sinusoidal phenomena, 186
slope of a line with trigonometry, 387
SOH-CAH-TOA, 376
solving right triangles, 379
solving trigonometric equations, 332
spring-mass system, 210
square formula, 8
SSA triangle, 399
SSS triangle, 399
stretching, 29
sum and difference formulas, 233
sum of cubes, 9
sum to product formulas, 271
supplementary angles, 371
symmetric domains, 38
symmetry, 35
symmetry of trig functions, 162
W
wrapping function, 118
Y
yearly precipitation, 186
yearly temperature, 186
T
tangent, 145, 376
tides, 186
trigonometric equations, 332
trigonometric functions, 145
trigonometric functions for right triangles, 376
U
unit circle, 114
V
verifying identities, 224
verifying inverse trigonometric identities, 319
verifying trigonometric identities, 225,
262
448