4.4 Evaluating Logarithms and change of Base Theorem
* common Logarithms
for all positive numbers x
Note:
If
if
if
* Natural Logarithm
for all positive numbers x
* Change of base Theorem
for any posit
7.6 Trigonometric equations:
A Chart for
special angles
Solving by linear methods:
Ex: solve
over
or
Solving by Quadratic methods:
Ex: Solve
over
or
No solution
Solving by trigonometric identities:
9.1 systems of linear equations
Linear system
A set of equations is called a system and the solutions of the system of equations must satisfy
every equation in the system If each equation in a system
7.5 Inverse circular functions
Inverse function
1. In a 11 function each x is related to only one y and each y is related to only one x
value
2. If f is a 11 function then f has an inverse function
9.2 Matrix solutions of linear systems
The Gauss Jordan method
Taking the coefficients and constants and arranging them in a matrix as follows
coefficients
constants
this is called the augmented matr
9.3 Determinant solution of Linear systems
Determinants:
Every nxn matrix A is associated with a real number called the determinant
of A written
1. Determinant of 2X2 matrix
If A =
Then
Ex: Let
find
9.5 Nonlinear systems of equations
1. .solving a system of two equations with two variables
with one of the equations is Linear
Ex: solve
solution:
take the linear equation and solve for x or for y
re
9.8 Matrix Inverses

The identity matrix
Ex: Let
show that
Solution:
Multiplicative inverses:
If A is an nxn matrix and
there is a matrix A such that
is called the inverse of matrix A
Finding the i
9.7 Properties of Matrices
a matrix (plural matrices) is a rectangular array of numbers, symbols, or
expressions, arranged in rows and columns.
The individual items in a matrix are called its elements
10.1 Parabolas
Definition:
AParabola is a set of points whose distance from a fixed point called (focus) equals its
distance from a fixed line called (directrix)
FM = MD
Screen clipping taken: 7/24/20
Reduction formula
* Functions of the form
this function can be written in the form
Where
Ex: Rewrite
in the form
Solution,
Ex: Graph
x
Chapter 7 Trigonometric identity and equations (2) Page 1
x
x+
y
10.2 Ellipses
Definition: An ellipse is the set of points in a plane the sum of whose distances from two
fixed points called (foci) is constant
horizontal ellipse
vertical ellipse
An ellipse centered
103 Hyperbolas
In the graph of each hyperbola considered so far, the center is the origin and the asymptotes pass
through the origin. This feature holds in general; the asymptotes of any hyperbola pa
7.7 Inverse trigonometric equations:
Ex: Solve
Ex: solve
Let
=
A is in 1st quadrant
Ex. solve
let
A is in 1st quadrant
Ex: solve
it has no solution
Ex: Solve.
Chapter 7 Trigonometric identity and equa
7.4 Double angle and half angle Identities
Double angle Identities
Ex: Given that
and
Find:
X is in 2nd quadrant
8
Chapter 7 Trigonometric identity and equations (2) Page 1
17
Ex: Find the values of t
Chapter 5
5.1
Angles:
An angle consists of two rays or two segments in a plane with a common end point
moves anticlockwise
moves clockwise
*The degree measure of an angle
1 rotation =360 degrees
* Typ
4.1 Inverse Functions
Onetoone Functions:
A function
i.e. each xvalue corresponds to only one yvalue, and each yvalue correspond to
only one Xvalue
Note: A onetoone function is either increasi
4.3 Logarithmic Functions
Definition:
For all real numbers y and all positive numbers
a and x where
if and only if
argument
base
EX: Solve each equation
1.
2.
3.
Definition:
If
and
then
defines the lo
5.3 Evaluating Trigonometric Functions:
* Cofunctions : Cofunction Identities:
In the adjacent figure write the missing angle in terms of A
its measure is:
A
So
these are called cofunctions
in the sam
5.2 Trigonometric Functions:
In aright angled triangle far an acute angle
(theta)
Hypotenuse
opposite
to
adjacent
to
Ex: Find the values of the six trigonometric functions of angle
passes through the
6.2 The Unit Circle and Circular Functions
Ex: Find the exact value of
1.
2.
3.
Note:
4.
5.
6.
7.
Note:
Conterminal =
reference =
8.
Note:
conterminal =
reference =
9.
Math002 Page 1
Ex. If
is the ref
5.4: Solving right triangle
Definition:
Solving aright triangle means finding all unknown angles and sides in the tri angle .
Ex: Solve right triangle ABC; if
and
where C is the right angle
Solution:
Chapter 6: the Circular Functions.
6.1: Radian Measure
Radian:
An angle with its vertex at the center of a circle that intercepts an arc on the
circle equal in length to the radius of the circle has a
6.3 Graphs of Sine and Cosine Functions
Periodic Function:
is a function
such that
for every real number x in the domain of
, every integer n, and some positive real number p
The least possible positi
6.4 Translations of the graph of sine and Cosine functions
Review:
Ex: Graph
over one period
solution:
Ex: Graph
over one period
solution:
Math002 Page 1
over two periods
Ex: Graph
solution:
over two
Chapter 7: Trigonometric identity and equations
71  Fundamental Identities:
Reciprocal Identities
_Quotient Identities
Pythagorean identities
Negative angle identities
Note:
and
Ex: If
and x is in 4