Give me an e i
!
Whats that spell? It is said that Euler had e i placed on his tombstone. (Recall this is part of Eulers
formula, since he was the first to develop it) He was noted to have a dry sense of humor, so this would
be fitting. Dont know if the
Students,
I am sure there are many ways people can talk about how you remember what trig functions are positive
and negative in which quadrants. I can think of several acronyms that people use to remember this but I am
not one to teach or learn by acro
There are several uses of trig identities. Among them, we can solve equations, simplify
expressions, and proof (verify) theorems.
Solving equations.
Generally, we solve equations in two stages:
a. Solve for one or more of the trig ratios. This may deal on
Characteristics of exponential and logarithmic functions
Families of functions.
1. Linear: y = mx + b
2. Quadratic : y = ax2 + bx +c
3. Polynomial: y = axn + bx n-1 + cx n-2 +
4. Power: y = x n
Exponential: y = b x
ogarithmic : x = b y so we define y = l
Operations on
matrices
A matrix is an ordered array of
information. The matrix is made of
with rows and columns. The dimension
of a matrix is the number of rows and
then the number of columns.
For example, A is a 2 row by 3 column
n 1
A
same thing with o
TRIGONOMETRY
Importance of calculator settings
MODE
IN CALCULATION
If we are in the wrong mode, our answers will be wrong!
Mode should agree with units in the problem.
Example: solve for x
sin 67 = 50/x
x =
50
/sin 67
x - 58.4
MODE
IN CALCULATION
If we ar
Solving Trigonometric Equations
The methods of solving these equations are not cut and dried no one approach will work on all problems.
However, there are some general rules that will help. These rules will also apply to simplifying trigonometric
expressi
TRIGONOMETRY
The trigonometry of right triangles
Coordinate trigonometry
RIGHT TRIANGLE TRIGONOMETRY
Definitions of the trig ratios:
sine
sin
opposite leg
hypotenuse
SOH
Note:
cosine
cos
adjacent leg
hypotenuse
CAH
tangent
tan
opposite leg
adjacent leg
Trigonometry
Angles and their measures
Angle measurement
1.
Degree measure ( 1 revolution = 360 )
a. Decimal degrees
b. Degrees minutes seconds
2.
(1 = 60 and 1 = 60 )
Radian measure ( 1 revolution = 2 radian)
a.
Definition: consider a central angle. A ra
Systems of linear equations
Solving using Matrices
Gauss Jordan elimination
Traditional methods of solving systems
Matrices
Consider the system:
3x y + z = 5
2x + 2y + 3z = 4
4x 2y z = 3
Consider the system:
3x y + z = 5
2x + 2y + 3z = 4
4x 2y z = 3
Befor
Using technology
Systems of Equations
And
Matrices
Technological methods of solving systems
Inverse matrices
An inverse matrix is one such that the product of a matrix and
its inverse yields the identity matrix. The identity matrix is a
square matrix with
EQUATIONS AND IDENTITIES
Solving trigonometric equations and using the
identities
TRIGONOMETRIC IDENTITIESRatio Identity
tan
sin
cos
Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
1 + cot2 = csc2
Sum and Difference Identities
sin ( ) = sin () c
Rewriting expressions and solving equations
The Basic Rules of Logs
1) (defn) y = log b x implies b y = x
2) Product rule: log b (XY) = log b X + log b Y
3) Quotient rule: log b (X/Y) = log b X log b Y
4) Power rule: log b X Y = Y log b X
5) Converting ba
Forms of complex numbers.
1. Rectangular form: (standard from) a + bi
(some texts use j instead of i)
2. Trigonometric form: (polar form in some texts)
r(cos + i sin ) or r cis for short.
3. Euler form: (exponential form) r e i
( must be in radians)
Arga
TRIGONOMETRY
COMPLEX
NUMBERS
AND
Forms of Complex numbers and their
conversions
RECTANGULAR VERSUS POLAR
COORDINATES
Consider any point in a plane,
RECTANGULAR VERSUS POLAR
COORDINATES
Consider any point in a plane,
How can we identify that point relative
The Rules of Logarithms.
First rule of logarithms.
(defn) y = log b x can be written b y = x
where b > 0 so x > 0
Exponential form: b y = x
Logarithmic form: y = log b x
Logs are exponents!
Example: estimate log 5 64
What power do I raise 5 to get 64?
WORKING WITH IDENTITIES
Special problems dealing with one angle,
Sum and Difference identities for two angles
TRIGONOMETRIC IDENTITIES
Basic Identities
TRIGONOMETRIC IDENTITIES
Basic Identities
TRIGONOMETRIC IDENTITIES
Basic Identities
TRIGONOMETRIC IDENT
I first changed the plotting
range to (-2, 6) for x and y
Graphing logs using MS Math
This program can graph equations in implicit (y = ) form like other
calculators/programs, but will also graph implicity (y = not required)
I then clicked on proportional
Solving parts of a right triangle.
We generally use these steps to solve.
1. Draw the right triangle and label any sides or angles that are known.
2. Using one of the acute angles, assign names to the sides. This is opposite leg (OL),
adjacent leg (AL) an
EQUATIONS AND IDENTITIES
Solving trigonometric equations and using the
identities
TRIGONOMETRIC IDENTITIESRatio Identity
tan
sin
cos
Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
1 + cot2 = csc2
Sum and Difference Identities
sin ( ) = sin () c
Angles and their measure
Converting Angle Measures
Radian Measure
Radians have no units! They are
just a real number. The measure
of an angle in radians is the
ratio of the arc length inside the
angle to the radius length.
Arc length:
a) Dec. Degrees to
d
Week Six
As in Week Six, this week we focused on one TCO:
5
Given a 2x2 or 3x3 systems of linear equations and unknowns, solve for the unknown variables
using algebraic techniques, matrix methods such as the inverse of the coefficient matrix and
Gaussian
Rules of exponents (E)
1. definition
b
n
1. definition
y
log b x = y means b = x
bbbb
=
Rules of logarithms (L)
b > 0 and b1,
y is any real, x > 0
with n factors
x
2. b b
3.
bx
b
y
y
b
x y
4. (b )
5. b
=
=
-1 1
= /b
b
x+y
2. log b (MN) = log b M + log b N
Systems of Linear Equations
Three Types of Systems
and
Traditional methods of solving
Types of solutions
A solution to a system is a set of values
that make all the equations true.
Consistent and independent yield only one solution
(in two unknowns, this
Course outline
Expectations
Grading
Attendance
In the course shell (accessed through DeVry login)
Course Home
Syllabus, Policies, Student Resources
Weekly Material
Announcements
Gradebook
Doc Sharing
In MML: (accessed through Assignments link each
week)
H
Angles
Definitions: Know the definitions introduced in this paragraph! Angles in standard position
(1), have the initial side as the positive x axis and the vertex of the angle at the origin. The
angle is then named by the quadrant in which the terminal
s
3
y=x
y 10x
2
y = 1/10 x 3
y=x
y = x
y=x
Inverse functions are symmetrical when viewed about the line y = x. If there is not a perfect symmetry,
the two functions are not inverses.
y=5x
y=5x
y = log(5x)
y=x
y=x
y = log5(x)
The first pair of graphs are not
Week Five
In week three, we focused on two special functions that are in fact inverses of each other.
The exponential functions are y = b x where b is positive but not equal to 1. When the base is
greater than one (red graph), the function is increasing t
TRIGONOMETRY
Special angles and the unit circle
SPECIAL ANGLES
Special Angles - we consider the quadrantal angles as
well as 30, 45, and 60
SPECIAL ANGLES
Getting the exact values for sin, cos, tan for 30 and 60
From Geometry
In an equilateral triangle, t