Trigonometry
5.1 Introduction
Trigonometry is the study of triangles. Triangles rather than, say, squares or
hexagons because any other polygon (a closed shape with straight edges) can
be constructed by adding triangles together (Fig. 5 .1 ). Thus, if the
1
Design and Implementation of a Pure Sine Wave Single Phase Inverter for
Photovoltaic Applications
1
Mohamed A.Ghalib1, Yasser S.Abdalla 2, R. M.Mostafa3
Automatic Control Department, Faculty of Industrial Education, Beni-suef University, Egypt.
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Engineering Handbook
We have been a leader in the rotary components
industry for over 50 years. Our staff includes electrical,
mechanical, manufacturing and software engineers,
metallurgists, chemists, ph
SOLUTIONS OF TRIANGLES
23
Lesson
Learning Outcomes and Assessment Standards
Learning Outcome 3: Shape, space and measurement
Assessment Standard
Solve problems in two dimensions by using the sine, cosine and area rules, and
by constructing and interpretin
How to Graph Trigonometric Functions
This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal
shifts, and vertical shifts of trigonometric functions.
The Unit Circle and the Values of Sine and Cosine Functions
The u
Creating A Sine Wave In Excel
Step 1.
Create Columns in Excel for: Frequency, Circular Frequency,
Omega (rad/s), Amplitude, Delta t, Time, and Sine Wave.
Step 2.
Enter Desired Values for Frequency, Omega, Amplitude, and
Delta t (sec.)
Step 3. Fill in Colu
MEP Pupil Text 4
4 Trigonometry
4.1 Squares and Triangles
A triangle is a geometric shape with three sides and three angles. Some of the different
types of triangles are described in this Unit.
A square is a four-sided geometric shape with all sides of eq
GCSE (91)
Transition Guide
J560
MATHEMATICS
Theme: Trigonometry
April 2016
Oxford Cambridge and RSA
GCSE (91)
MATHEMATICS
We will inform centres about any changes to the specification. We will also
publish changes on our website. The latest version of our
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 28
M2
GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine
Student Outcomes
Students use graphing calculators to find the values of sin and cos for between 0 and 90.
Students solve for missing sides of a
Lecture 45b
Ellipse
Quadric curves can be defined geometrically in terms of distances between
points.
In particular, using two nails , a rope
and a pencil we can draw an ellipse:
Note : d1 + d2 is the same for all points on the ellipse. It is simply the
l
Lecture 45e Theory Questions
1. Based upon the conditions provided for each part, classify the curves that satisfy the
equation, 2 + 2 + = 0:
a. = > 0, < 0
b. > 0, < 0, < 0
c. < 0, < 0, < 0
d. = 0
e. = 0
(Please continue with the Lecture 45e Problems on t
Lecture 45
Cramers Rule
(
Ex : Consider the system
to eliminate y :
(
ax + by = r d
ax + by = r
cx + dy = s
(
=
and apply elimination procedure
adx + bdy = rd
cx + dy = s (b)
cbx bdy = bs
a
r b
c
s d
rd bs
. Likewise, y =
=
= x =
a b
a
ad bc
c d
c
C
Lecture 39 Theory Questions
1. For each of the reduced row echelon form of the augmented matrix of a system below,
a. State the number of solutions of the system
b. State the solution(s), if any.
1
i. [0
0
1
ii. [0
0
1
iii. [0
0
1
iv. [0
0
0
1
0
0
0
0
0
0
Lecture 45d Theory Questions
1. Describe how the graphs of the hyperbolas 2 2 = 1 and 2 2 = 1 would differ.
2
2. Suppose the hyperbola 2
2
1
= 1 has a large value for a. Describe how the graph
would appear.
3. Write an equation of a hyperbola with the fo
Lecture 42 Theory Questions
1. Let A be a 2x3 matrix and B be a 3x4 matrix. Explain why .
2. Does each of the following matrices have an inverse? If so, find its inverse and verify
that
1 = = 1 .
2
a. [
1
2
b. [
3
5
]
3
4
]
6
3. Does the identify matrix
Lecture 35 Theory Questions
1. Use the definition of exponential equations to determine whether each of the following
is an exponential equation. In each case, explain your reasoning.
a. 2 = 9
b. 2 = 7
c. 2 = 3
d. 2 + 6
e. = 3
f.
25 =
g. () = 3
2. Find t
Lecture 37 Theory Questions
1. Which of the following systems of equations are linear?
a. 2 + 2 = 2
+ =2
b.
1
+ = 2
+ =3
c. + 2 = 4
=2
d. 23+4 + = 0
+ =1
2. True or False. Explain your answer.
a.
A system of two linear equations can have no solution.
b.
Lecture 32 Theory Questions
1) As + and as 0. Use this to answer the following:
a) as +
i) ?
ii) (1) ?
iii) + 1 ?
b) as
i) ?
ii) (1) ?
iii) + 1 ?
2) The graph of () has a horizontal asymptote = 5.
a) What is the horizontal asymptote of the graph of () +
Lecture 45d
Hyperbola
Hyperbolas are obtained, as with ellipses, by fixing two points (foci ) and by
imposing the condition
d1 d2 = l |d1 d2 | = l
where l is a constant. (See the figures.)
Case 1. The foci are on the X -axis.
|d1 d2 | = l
( l = 2a , see t
Lecture 38 Theory Questions
1. By definition, which of the following are matrices?
a. [1
2 3]
1
b. [2]
3
1
c. [
4
1
d. [
2 3
]
5 6
2 3
]
4
5
1 2 3
e. [ 4 5 ]
6
2. Suppose that a system of equations consists of four linear equations with five variables.
Wh
Lecture 33 Theory Questions
1) Assume that and 1 exist.
a) If (2) = 3 what is 1 (3) = ?
b) if () = what does 1 () = ?
c) if 2 = what is 2 () = ?
2) Suppose that point (2,3) is on the graph = . Find a point on the graph of
().
3) How are the asymptotes of
Lecture 40 Theory Questions
1. True or False: A 2x2 matrix can be equal to a 3x3 matrix. Explain your reasoning.
2. True or False: A 2x2 matrix and a 3x3 matrix can be added together. Explain your
reasoning.
3. Let A be a 5x6 matrix with 32 = 5 and B be a
Lecture 36a Theory Questions
1. Suppose that a quantity triples for every time interval t. Is this an example of
exponential growth? (Hint: replace 2 with 3 and repeat the derivation found in
lecture #36a).
2. Does the equation, () = (0) , describe expone
Lecture 43
(
Matrix Equations
x + 2y = 5
1 2
This can be written in matrix form. Let A =
3x y = 1
3 1
x
5
be the coefficient matrix of the system and X =
, B=
. Then the
y
1
system is equivalent to the single matrix equation
AX = B
1
1
Suppose we know
Lecture 41 Theory Questions
1. Let the dimension of A be 3x1, the dimension of B be 3x2, the dimension of C be 3x3 and
the dimension of D be 2x3.
Which of the following products are defined? Explain.
a.
b.
c.
d.
e.
f.
g.
2. Let = [
0 0
2 3
], B= [
]
Lecture 45 Theory Questions
1. Explain why Cramers rule is only applicable to a system where the number of equations
is equal to the number of variables.
2. Is Cramers rule applicable if the determinant of the system is equal to 0? Explain.
3. Consider th
Lecture 45b Theory Questions
1. Describe how an ellipse would look like if its eccentricity is 0.
2. Describe how an ellipse would look like if its eccentricity is close to 1.
3. If the foci of an ellipse are on the y-axis, which semi-axis of the ellipse
Lecture 45f Theory Questions
1. Rate the following statements as True or False. Provide an example or a counterexample
(Hint: You can use graphs).
a. A system of non-linear equations cannot have exactly two solutions.
b. A system of non-linear equations c
Lecture 44 Theory Questions
1. Suppose that the first row in a square matrix consists of zeros. What is the determinant
of the matrix?
2. What is the determinant of an identity matrix?
3. Consider the following diagonal matrix:
1
[0
0
0
0
2
0
0
0
0
3
0
0
Lecture 43 Theory Questions
1. Consider the system of equations below:
2 + 2 = 4
3 4 = 1
Find A, B, and X to rewrite the system as a single matrix equation AX = B
A=
B=
X=
2. Let = [
1 2
3
], = [ ], X= []
3 3
5
Write the matrix equation AX = B as a system