Trigonometry_150627_01

© Art Traynor 2011
Mathematics
Definitions
Expression
Formula
A concise symbolic
expression positing a
relationship between
quantities
Function
A Relation between a Set of
inputs and a Set of permis-
sible outputs whereby each
input is assigned to exactly
one output
Equation
A symbolic formula, in the
form of a proposition,
expressing an equality
relationship
Proposition
A declarative expression
asserting a fact whose truth
value can be ascertained
Univariate: an equation containing
only one variable
Multivariate: an equation containing
more than one variable
(e.g. Polynomial)
Linear Equation
An equation in which each term is either a constant or the
product of a constant and (a) variable[s] of the first order
Symbols
VariablesConstants
Designate expression
elements consisting of:
Functions
Definitions – no
Page # citation
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
A finite combination of symbols, of discrete arity,
that is well-formed
Operands ( Terms )
Operations
Transformations capable of
rendering an equation or
system of equations into
equivalence
As in a differential (e.g.
Differential Equation)
As in a coefficient
Trigonometric Functions

© Art Traynor 2011
Mathematics
Expression
Mathematical Expression
Algebra
A Mathematical Expression is a precursive finite composition
to a Mathematical Statement or Proposition
( e.g. Equation) consisting of:

a finite combination of Symbols
possessing discrete Arity
Expression
A finite combination of symbols,
of discrete arity,
that is well-formed
that is Well-Formed

© Art Traynor 2011
Mathematics
Arity
Arity
Expression
The enumeration of discrete symbolic elements ( Operands )
comprising a Mathematical Expression is defined as
its Arity

The Arity of an Expression is represented by
a non-negative integer index variable ( ℤ + or ℕ ),
conventionally “ n ”

A Constant ( Airty n = 0 , index ℕ )or Nullary represents
a term that accepts no Argument

A Unary or Monomial has Airty n = 1
VariablesConstants
Operands
Expression
A relation can not be defined for
Expressions of arity less than
two: n < 2
A Binary or Binomial has Airty n = 2
All expressions possessing Airty n > 1 are Polynomial
also n-ary, Multary, Multiary, or Polyadic


© Art Traynor 2011
Mathematics
Arity
Arity
Expression
VariablesConstants
Operands
Expression
A relation can not be defined for
Expressions of arity less than
two: n < 2
Nullary
Unary
n = 0
n = 1 Monomial
Binary n = 2 Binomial
Ternary n = 3 Trinomial
1-ary
2-ary
3-ary
Quaternary n = 4 Quadranomial4-ary
Quinary n = 5 5-ary
Senary n = 6 6-ary
Septenary n = 7 7-ary
Octary n = 8 8-ary
Nonary n = 9 9-ary
n-ary

© Art Traynor 2011
Mathematics
Equation
Equation
Expression
An Equation is a statement or Proposition
( aka Formula ) purporting to express
an equivalency relation between two Expressions :

Expression
Proposition
A declarative expression
asserting a fact whose truth
value can be ascertained
Equation
A symbolic formula, in
the form of a proposition,
expressing an equality
relationship
Formula
A concise symbolic
expression positing a
relationship between
quantities
VariablesConstants
Operands
Symbols
Operations
The Equation is composed of Operand terms and
one or more discrete Transformations ( Operations )
which can render the statement true

© Art Traynor 2011
Mathematics
Linear Equation
Linear Equation
Algebra
An Equation consisting of:
Operands that are either
Any Variables are restricted to the First Order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
n Constant(s) or
n A product of Constant(s) and
one or more Variable(s)
The Linear character of the Equation derives from the
geometry of its graph which is a line in the R2 plane

As a Relation the Arity of a Linear Equation must be
at least two, or n ≥ 2 , or a Binomial or greater Polynomial


© Art Traynor 2011
Mathematics
Linear Equation
Equation
Standard Form ( Polynomial )
 Ax + By = C
 Ax1 + By1 = C
For the equation to describe a line ( no curvature )
the variable indices must equal one

 ai xi + ai+1 xi+1 …+ an – 1 xn –1 + an xn = b
 ai xi
1 + ai+1 x 1 …+ an – 1 x 1 + a1 x 1 = bi+1 n – 1 n n
ℝ
2
: a1 x + a2 y = b
ℝ
3
: a1 x + a2 y + a3 z = b
Blitzer, Section 3.2, (Pg. 226)
Section 1.1, (Pg. 2)
Test for Linearity
 A Linear Equation can be expressed in Standard Form
As a species of Polynomial , a Linear Equation
can be expressed in Standard Form
 Every Variable term must be of precise order n = 1

© Art Traynor 2011
Mathematics
Definition
Unit Circle/Standard Position Definitions
A Unit Circle is formed by superimposing a circle of radius = 1 over a rectangular coordinate system
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ
 Ray/Line Segment l1, formed of points ‘O’ (0,0),
the origin, and ‘A’ (1, 0), (the “initial side”) is
rotated counterclockwise (positive direction) to a
point ‘P’ (x, y)
Terminal Side
Initial Side
t
θ = ‘ t ’ radians
 Ray/Line Segments l1 and l2, forming an angle
AOP whose measure is θ
 “Standard Position ” is defined as an angle so
situated with its vertex at the origin and its initial
side aligned on the positive x-axis

© Art Traynor 2011
Mathematics
Definition
Angular Measures
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ = 1
 Degrees:
Terminal
Side
Initial Side
t = 1
 1° = 1/360th of a complete revolution about
the unit circle
n One Minute ( 1′ ) is 1/60th of a Degree
n One Second ( 1″ ) is 1/60th of a Minute
 Radians:
 A standard unit of angular measure wherein
arc ‘AP ’ is identical in length to the initial
side ( line segment/ray ‘OA ’ ) of AOP
superimposed over the unit circle
n OA = OP = AP = t = θ = 1

© Art Traynor 2011
Mathematics
Definition
Angular Measures
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ = 1
 Degrees:
Terminal
Side
Initial Side
t = 1
the unit circle
 History – Origin of Convention
n Undetermined: Ancient Persian calendar
evidence suggests the days of the year were
numbered at three-hundred sixty ( 360 )
o Coincides with ancient astronomical
observation correlating the progress of the
sun’s rising to 1° per day
o Also has a correspondence with the
ancient world of sexagesimal (base 60)
number systems.

© Art Traynor 2011
Mathematics
Definition
Angular Measures
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ = 1
 Degrees:
Terminal
Side
Initial Side
t = 1
the unit circle
n Undetermined:
o Progression of the Sun’s Rising
o Sexagesimal Numeration Correspondence.
 Still used in modern times in measuring
angles, geographic coordinates, and time
 Came to the Greek world by way of
the Babylonians

© Art Traynor 2011
Mathematics
Definition
Angular Measures
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ = 1
 Degrees:
Terminal
Side
Initial Side
t = 1
the unit circle
n Undetermined:
 Babylonian geometers partitioned a
circle into precisely six sectors defined by
by the Equilateral Chord Triangle
( which was then subpartitioned into 60
equal sub-intervals )

© Art Traynor 2011
Mathematics
Definition
Angular Measures
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ = 1
 Degrees:
Terminal
Side
Initial Side
t = 1
the unit circle
n Undetermined:
 The number sixty also ( coincidentally )
enjoys some highly useful properties
 Superior Highly Composite Number
 Has 12 factors :
{ 1, 2, 3, 4, 5, 6,
10, 12, 15, 20, 30, 60 }

© Art Traynor 2011
Mathematics
Definition
Angular Measures
x
y
A (1, 0)
P (x, y)
l1
l2
O
U
θ = 1
 Degrees:
Terminal
Side
Initial Side
t = 1
the unit circle
n Undetermined:
 The number sixty also ( coincidentally )
enjoys some highly useful properties
 Has Three Prime Factors { 2, 3, 5 }
 Particularly convenient for time :
{ 30-min, 20-min, 15-min,
12-min, 10-min, 6:1-min }

© Art Traynor 2011
Mathematics
Definition
Angular Measures - Radians
x
y
A ( 1, 0 )P ( -1, 0 )
l1l2
O
U
θ = 180 °
Terminal Side Initial Side
t = π θ = ‘ t ’ radians

When the terminal side of AOP, ( line
segment/ray ‘OP ’ ) is rotated halfway through
the unit circle, ‘P ’ becomes aligned with the x-
axis at ( -1, 0 ), θ = 180°

Circumference of a Circle: 2πr implies
that ½ the circumference of a circle is πr ,
and superimposed on the unit circle:
OA = OP = r =1
AP = t = π
180° = π radians
π
180
1° =




 1 radian =
180
π( )
°
≈ 57.29578°
≈ 0.01745 radians

© Art Traynor 2011
Mathematics
Definition
Angular Values – Common Angles
x
y
U
0 ° :: 0
Π
6
30 ° ::
Π
4
45 ° ::
Π
3
60 ° ::
Π
2
90 ° ::
2Π
3
120 ° ::
3Π
4
135 ° ::
5Π
6
150 ° ::
180 ° :: Π
7Π
6
210 ° ::
5Π
4
225 ° ::
4Π
3
240 ° ::
3Π
2
270 ° ::
5Π
3
300 ° ::
7Π
4
315 ° ::
11Π
6
330 ° ::
360 ° :: 2Π
Twos Sixes Fours Threes
30 °
45 °
60 °
90 °
120 °
135 °
150 °
180 °
210 °
225 °
240 °
270 °
300 °
315 °
330 °
360 °
Π
6
Π
4
Π
3
Π
2
2Π
3
3Π
4
5Π
6
Π
7Π
6
5Π
4
4Π
3
3Π
2
5Π
3
7Π
4
11Π
6
2Π

© Art Traynor 2011
Mathematics
Definition
Central Angle Properties
x
y
A
B
O
U
θ
 Length of a Circular Arc:
 Where an arc of length s on a circle of
radius r subtends a central angle of radian
measure θ, then
n s = r θ
r
A Central Angle of a circle is an angle θ whose vertex is positioned at the center of the circle such
that the arc bounding the circular sector ‘AB ’ is said to Subtend θ , such that:
s
 Area of a Circular Arc:
 Where θ is the measure of a central angle of
a circle of radius r , then the area A of the
circular sector determined by θ is given by
n A = ½r 2 θ

© Art Traynor 2011
Mathematics
Equality
Unit Circle
Properties of Triangles
x
y
O
U
Begin with the Unit Circle
of Radius = 1
①
①
r = 1
② Introduce Line Segments
(Chords) between each of
the points of intersection
with the Unit Circle and
the coordinate axes
②
②
②
a Note this forms a square
b The diagonal of the
square is equal to
r + r = 2r or 1 + 1 = 2
c Note that the top half of
this square forms two
equal (right) triangles
2
d By the Pythagorean
theorem, the length of the
Chords are 12 + 12

© Art Traynor 2011
Mathematics
Equality
Unit Circle
x
y
O
U r = 1
2
③ Any one of the three Rigid
Motion Transformations, can
be applied to demonstrate
Equality ( one triangle can be
superimposed on any other )
a Rotation
b Translation
c Reflection

© Art Traynor 2011
Mathematics
Transformation
Transformation
A Transformation is a species of FMM
(Function~Map~Morphology) which
Maps Elements ( as of a Set ) of one Metric Space to
another Metric Space ( or to a Subspace of itself )

FMM
Rotations is well-defined for all
Inner Product (Sub) Spaces
Active or Alibi Transformation:
A Domain Element of the Transformation Set
is assigned a to a non-identity CoDomain Element

Species:
Passive or Alias Transformation:
The Domain Element of the Transformation Set
is an identity to its CoDomain Element with the
Transformation operating on the coordinate system
effecting a change in Basis


© Art Traynor 2011
Mathematics
Transformation
Transformation
A Transformation is a species of FMM
(Function~Map~Morphology) which
Maps Elements ( as of a Set ) of one Metric Space to

FMM
Scalar Multiple: Expansion if the scalar is positive
Dilation if the scalar is negative

Species:
Rotation: Points constituting an axis remain invariant while all other
points are mapped through an angular Displacement

Reflection: Points along a unique intersection of the Domain form an axis
about which multiplicative inverses of the remaining Domain points
are mapped to a Codomain scaled by their distance from the axis

Translation: A linear Displacement
Shear: Points constituting one or more axes remain invariant while all
other points are subject to a Translation parallel to those axes and
proportional to their perpendicular distance from the axes


© Art Traynor 2011
Mathematics
Isometry
Isometry
A Transformation which Maps Elements ( as of a Set )
of one Metric Space to
such that all Elements ( e.g. points ) of the respective spaces
remain equidistant

Transformation
Motion “progressing through Length”
Direct Isometry: Translation
Indirect Isometry: Reflection
n Orientation Preserving
The Mathematical Object so transformed
by an Isometry thus possesses the property of Congruence
with its Codomain Transform Map
Isometry is a test for Congruence
The Set of all Congruences for an
Object is the Union of all its well-
defined Isometries
Two Objects are Congruent if they
are related by an Isometry
Composite Isometry: one or more isometries Mapped in sequence
Deformations such as Folding,
Cutting, Melting, Bending, Stretching,
and Twisting are not considered
Isometries
A Bijective Map

© Art Traynor 2011
Mathematics
Rotation
Rotation
A Rotation is a Rigid Body motion
which maps the points of a Mathematical Object
( or coordinate system) in an orientation-preserving orthogonal
Transformation through an angular Displacement

Transformation
Any Composite Isometry ( of period p > 2,
i.e. two or more successive sequences of Transformation)
is either a Rotation or Reflection

Properties:

© Art Traynor 2011
Mathematics
Equality
Unit Circle
x
y
U r = 1
2
a Rotation
b Translation
c Reflection
2
O

© Art Traynor 2011
Mathematics
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
2
O
r = 1

© Art Traynor 2011
Mathematics
2
r = 1
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
O
r = 1

© Art Traynor 2011
Mathematics
r = 1
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
2
O
r = 1

© Art Traynor 2011
Mathematics
r = 1
O
2
r = 1
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
O
r = 1

© Art Traynor 2011
Mathematics
O
2
r = 1
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
O
r = 1
2

© Art Traynor 2011
Mathematics
r = 1
r = 1
O
2
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
O

© Art Traynor 2011
Mathematics
r = 1
r = 1
O
2
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
O
r = 1
2

© Art Traynor 2011
Mathematics
r = 1
Equality
Unit Circle
x
y
U
2
a Rotation
b Translation
c Reflection
2
O

© Art Traynor 2011
Mathematics
Reflection
Transformation
Reflection
A Transformation in which one Element ( point ) of the
Reflection Set is intersected by an n – 1 Ambient Space axis
( Hyperplane ) with the Reflective Codomain consisting of the
multiplicative inverses of each remaining point of the Domain scaled
by their distance from the Hyperplane axis
 Rotations is well-defined for all
Every Isometry in a plane is the product of at most three
Reflections, at most two if the Mathematical
Object under Transformation features a fixed point

Properties:
Chiral: The reflection is a mirror image or can be superimposed
upon its preimage via a simple Translation

Species:
Amphichiral: The reflection image cannot be superimposed upon
its preimage via a simple Translation


© Art Traynor 2011
Mathematics
Fixed Point
Transformation
Fixed Point
A Fixed Point is an Element
for which a Function will Map the identical element
in both Domain and CoDomain Sets
 Also known as an Invariant Point
f(c) = c
Examples:
y = x
{ 2 } for f(x) = x2 – 3x + 4
f(x) = x + 1 has no fixed point as x ≠ x +1

© Art Traynor 2011
Mathematics
Hyperplane
Inner Product Space ( IPS )
An Inner Product Space
is a Vector Space
over a Field of Scalars (e.g. R or C )
Structured by an Inner Product

These spaces have a well-ordered semantic construction of the form:
“ The IPS of conventional multiplication over the field of R ”
“ The IPS of the dot product over the field of R ”
Transformation

© Art Traynor 2011
Mathematics
Definition
Trigonometric Functions – Acute Angles
x
y
O
U
θ
a (adj )
b (opp )
c (hyp )
sin θ =
b
c
opp
hyp( )
cos θ =
a
c
adj
hyp( )
tan θ =
b
a
opp
adj( )
csc θ =
c
b
hyp
opp( )
sec θ =
c
a
hyp
adj( )
cot θ =
a
b
adj
opp( )
Think “A” for Adjacent, and then b=opp
simply follows by process of elimination
*
Sin is the “fundamental” relationship
from which all others can “key”
*
θ is what we’re interested in, and the
opposite angle (b) is the most conspicuous
*
Cos just looks at the
“other” side ratio (to
the hypotenuse
*
The primacy of Sin
and of the opp angle
is echoed in tan where
opp is the numerator
*
tan θ = sin θ
cos θ( ) 1
opp/hyp( )
Sine is Prime and that’s why it Rhymes*
“A” is ayyyydjacent…*
It’s obeeevious that “B” is opposite*

© Art Traynor 2011
Mathematics
Definition
Trigonometric Functions – Obtuse (any) Angle
x
y
O
U
sin θ =
b
r
opp
hyp( )
cos θ =
a
r
adj
hyp( )
tan θ =
b
a
opp
adj( )
csc θ =
r
b
hyp
opp( )
sec θ =
r
a
hyp
adj( )
cot θ =
a
b
adj
opp( )P ( a, b )
Terminal
Side
θ
Initial Sidea (adj )
b (opp )
r (hyp )
Think “A” for Adjacent, and then b=opp
simply follows by process of elimination
*
Sin is the “fundamental” relationship
from which all others can “key”
*
θ is what we’re interested in, and the
opposite angle (b) is the most conspicuous
*
Cos just looks at the
“other” side ratio (to
the hypotenuse
*
The primacy of Sin
and of the opp angle
is echoed in tan where
opp is the numerator
*

© Art Traynor 2011
Mathematics
Definition
Inverse Trigonometric Functions
sin θ =
b
c
opp
hyp( ) sin-1 θ = arcsin θ
x
y
O
θ
a (adj )
b (opp )
c (hyp )
 Many of the trigonometric functions are not one-to-one, and thus do not have defined
inverse functions over their unrestricted domains
Injunction or Injective Function (one-to-one) where "a"b ( f(a) = f(b) a = b )
Every x A maps to B only once , and every f(x) B maps to A only once
Every f(x) B maps to A only once



 Inverse trig functions therefore must be defined
over appropriately restricted ranges
cos θ =
a
c
adj
hyp( ) cos-1 θ = arccos θ
tan θ =
b
a
opp
adj( ) tan-1 θ = arctan θ

© Art Traynor 2011
Mathematics
Definition
Inverse Trigonometric Functions
sin θ =
b
c
opp
hyp( ) sin-1 θ = arcsin θ
x
y
O
θ
a (adj )
b (opp )
c (hyp )
 Many of the trigonometric functions are not one-to-one, and thus do not have defined
inverse functions over their unrestricted domains
Injunction or Injective Function (one-to-one) where "a"b ( f(a) = f(b) a = b )
Every x A maps to B only once , and every f(x) B maps to A only once
Every f(x) B maps to A only once



 Inverse trig functions therefore must be defined over appropriately restricted domains
cos θ =
a
c
adj
hyp( ) cos-1 θ = arccos θ
tan θ =
b
a
opp
adj( ) tan-1 θ = arctan θ
sin-1( sin θ ) = θ
cos-1( cos θ ) = θ
tan-1( tan θ ) = θ
(g ○ f)(θ) = θf(θ) f -1(θ) = g(θ)

© Art Traynor 2011
Mathematics
Definition
Trigonometric Identities
Obtuse (any) Angle
x
y
O
U
P ( a, b )
Terminal
Side
θ
Initial Sidea (adj )
b (opp )
r (hyp )
csc θ =
r
b
hyp
opp( )= 1
sin θ( )= 1
b/r( )= 1
opp/hyp( )
sec θ =
r
a
hyp
adj( )= 1
cos θ( )= 1
a/r( )= 1
adj/hyp( )
cot θ =
a
b
adj
opp( )= 1
tan θ( )= 1
b/a( )= 1
opp/adj( )
cot θ =
a
b
adj
opp( )= cos θ
sin θ( )= a/r
b/r( )=
adj/hyp
opp/hyp( )
tan θ =
b
a
opp
adj( )= sin θ
cos θ( )= b/r
a/r( )=
opp/hyp
adj/hyp( )
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec 2 θ
1 + cot 2 θ = csc 2 θ
aka, the Pythagorean Relation
of trigonometry
These may have to be memorized!
Appeared in problem set 1.1 #15
*

© Art Traynor 2011
Mathematics
Rules of Differentiation – Functions (Trigonometric)
Where the Differential Operator is denoted as Dx and where Dx f(x) = f´(x)
Dx sin x = cos x
Dx tan x = sec 2 x
Dx cos x = - sin x
Dx cot x = - csc 2 x
Dx sec x = sec x tan x
Dx csc x = - csc x cot x
Easy
1
The Dx of tan is what we seec (twice) not coco
not coco co
Whenever CSC
shows up, it’s
always a negative
experience
2 3
Derivatives of
themselves +
4
only squares
Differentiation

© Art Traynor 2011
Mathematics
Rules of Differentiation – Functions (Trigonometric)
Where the Differential Operator is denoted as Dx and where Dx f(x) = f´(x)
Differentiation
sin2 θ + cos2 θ = 1
csc2 θ – cot2 θ = 1
sec2 θ – tan2 θ = 1
–
1
sin2 θ = 1sin2 θ
cos2 θ
–
1
cos2 θ = 1cos2 θ
sin2 θ
I’m pretty sure the title of this slide is
wrong…these appear to be identities
and not derivatives
*

© Art Traynor 2011
Mathematics
Definition
Angular Values – Common Angles
x
y
U
Π
6
30 ° ::
Π
4
45 ° ::
Π
3
60 ° ::
Π
2
Twos Sixes Fours Threes
30 °
45 °
60 °
90 °
120 °
135 °
150 °
180 °
210 °
225 °
240 °
270 °
300 °
315 °
330 °
360 °
Π
6
Π
4
Π
3
Π
2
2Π
3
3Π
4
5Π
6
Π
7Π
6
5Π
4
4Π
3
3Π
2
5Π
3
7Π
4
11Π
6
2Π

© Art Traynor 2011
Mathematics
Definition
Trig Function Values – Common Angles
x
y
Π
6
30 ° ::
Π
4
45 ° ::
Π
3
60 ° ::
θ
θ
1
√¯
3
2
2 √¯
2
1
Π
6
30 ° ::
1
2
√¯
3
2
a
b
r = c
sin θ
opp
hyp( )
b
r
cos θ
adj
hyp( )
a
r
tan θ
opp
adj( )
b
a
cot θ
adj
opp( )
a
b
Π
4
45 ° :: 1
Π
3
60 ° ::
sin-1θ = csc θ = 1
sin θ( )=
1
opp/hyp( )
√¯
2
2
√¯
3
2
cos-1θ = sec θ = 1
cos θ( )=
1
adj/hyp( )
√¯
2
2
1
2
√¯
3
3
√¯
3
cos-1θ = sec θ = 1
cos θ( )=
1
adj/hyp( )
√¯
3 11
θ
© Art Traynor 2011
csc θ
1
opp/hyp( )
csc θ
1
adj/hyp( )
√¯
3
1
√¯
3
3
2√¯
3
3
√¯
2
2
2
√¯
2
2√¯
3
3
1
b/r
1
a/r

© Art Traynor 2011
Mathematics
Definition
x
y
O
U
θ
a (adj )
b (opp )
r = c (hyp )
M A (1, 0)
P ( cos θ, sin θ )
1
sin θ
cos θ
lim sin θ = 0
θ→0
b
r
opp
hyp( )=
b
1
=
lim cos θ = 1
θ→0
a
r
adj
hyp( )=
a
1
=

© Art Traynor 2011
Mathematics
Differentiation
Derivatives of the Trigonometric Functions
Where the Differential Operator is denoted as D and where D f(x) = f (x), and where
x x
´
x denotes a real number of the radian measure of an angle
 Dx sin x = cos x

 Dx tan x = sec2 x
Dx cos x = - sin x
 Dx cot x = - csc 2 x

 Dx sec x = sec x tan x
*

© Art Traynor 2011
Mathematics
Differentiation
Derivatives of the Trigonometric Functions
Where the Differential Operator is denoted as D and where D f(x) = f (x), and where
x x
´
x denotes a real number of the radian measure of an angle
Dx sin x = cos x
Dx tan x = sec2 x
Dx cos x = - sin x
Dx cot x = - csc 2 x
Dx sec x = sec x tan x
*

© Art Traynor 2011
Mathematics
Definition
x
y
O
U
θ
a (adj )
b (opp )
r = c (hyp )
M A (1, 0)
1
tan θ
cos θ
Q
sin θ
 Area of a Triangle:
 Given triangle ∆ ABC (three-sided polygon,
composed of three line segments and three
opposing vertices) of height “ h”, the area is
given by
n A = ½bh
h
b
ac
A
B
C
 Area of a Circular Arc (Sector):
 Where θ is the measure of a central angle of
a circle of radius r , then the area A of the
circular sector determined by θ is given by
n A = ½r 2 θ
θ
r
r
Sandwich Theorem Application/Proof
s

© Art Traynor 2011
Mathematics
Definition
x
y
O
U
θ
a (adj )
b (opp )
r = c (hyp )
M A (1, 0)
1
tan θ
cos θ
Q
sin θ
 Area of a Triangle:
 A = ½bh
h
b
ac
A
B
C
 Area of a Circular Arc (Sector):
 A = ½r 2 θ
θ
r
r
MP = sin θ
AQ = tan θ
∆ AOP
Area of
< Sector AOP
Area of
< ∆ AOQ
Area of
½(1)MP
½bh
< <
½(1)sinθ
½(1) 2 θ
½r 2 θ
½(1)MQ
½bh
½(1)tanθ
< <½ sinθ ½θ ½ tanθ
Sandwich Theorem Application/Proof
s

Trigonometry_150627_01

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