This document discusses solving trigonometric equations graphically. It provides examples of using the intersection method and x-intercept method to find approximate solutions to trig equations by graphing them. Key steps outlined are writing the equation in the form f(x)=0, determining the period, graphing over the interval, and finding solutions graphically. It notes that trig equation solutions can also be expressed in degrees. Examples are provided for solving equations like sin(x)=a and tan(x)=b graphically in both radians and degrees.
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
Detailed Lesson Plan (ENGLISH, MATH, SCIENCE, FILIPINO)Junnie Salud
Thanks everybody! The lesson plans presented were actually outdated and can still be improved. I was also a college student when I did these. There were minor errors but the important thing is, the structure and flow of activities (for an hour-long class) are included here. I appreciate all of your comments! Please like my fan page on facebook search for JUNNIE SALUD.
*The detailed LP for English is from Ms. Juliana Patricia Tenzasas. I just revised it a little.
For questions about education-related matters, you can directly email me at mr_junniesalud@yahoo.com
Directions Please show all of your work for each problem.If.docxkimberly691
Directions
: Please show all of your work for each problem.
If applicable, you may find Microsoft Word’s equation editor helpful in creating mathematical expressions in Word.
There is a tutorial on using this equation editor in Module 1 Lecture Notes.
You also have the option of hand writing your work and scanning it.
1.
Find the greatest common factor.
4, 6, 12.
2.
Factor.
24
x
3
+ 30
x
2
3.
Factor out the GCF with a negative coefficient.
–24
m
2
n
6
– 8
mn
5
– 32
n
4
4.
Factor completely by factoring out any common factors and then factoring by grouping.
6
x
2
– 5
xy
+ 6
x
– 5
y
5.
The GCF of 15
y
+ 20 is 5.
The GCF of 15
y
+ 21 is 3.
Find the GCF of the product (15
y
+ 20)(15
y
+ 21).
6.
The area of a rectangle of length
x
is given by 15
x
–
x
2
.
Find the width of the rectangle in terms of
x
.
7.
Factor the trinomial completely.
x
2
+ 8
x
– 9
8.
Factor the trinomial completely.
2
x
2
+ 16
x
+ 32
9.
Complete the following statement.
6
a
2
– 5
a
+ 1 = (3
a
– 1)(__
?
__)
10.
State whether the following is true or false.
x
2
– 7
x
– 30 = (
x
+ 3)(
x
– 10)
11.
Factor completely.
x
2
+ 11
x
+ 28
12.
Factor completely.
15
x
2
+ 23
x
+ 4
13.
Factor completely.
6
z
3
– 27
z
2
+ 12
z
14.
The number of hot dogs sold at the concession stand during each hour
ii
h
after opening at a soccer tournament is given by the polynomial 2
h
2
– 19
h
+ 24.
Write this polynomial in factored form.
15.
Find a positive value for
k
for which the polynomial can be factored.
x
2
–
kx
+ 29
16.
Factor completely.
9
x
2
+ 4
17.
Determine whether the following trinomial is a perfect square.
If it is, factor the binomial.
x
2
– 12
x
+ 36
18.
Factor completely.
25
x
2
+ 40
xy
+ 16
y
2
19.
Factor.
s
2
(
t
–
u
) – 9
t
2
(
t
–
u
)
20.
State which method should be applied as the first step for factoring the polynomial.
6
x
3
+ 9
x
21.
State which method should be applied as the first step for factoring the polynomial.
2
a
2
+ 9
a
+ 10
22.
Solve the quadratic equation.
5
x
2
+ 17
x =
–6
23.
Solve the quadratic equation.
3
x
(2
x
– 15) = –84
24.
The sum of an integer and its square is 30.
Find the integer.
25.
If the sides of a square are decreased by 3 cm, the area is decreased by 81 cm
2
.
What were the dimensions of the original square?
26.
Write in simplest form.
27.
Write in simplest form.
28.
Write the expression in simplest form.
29.
The area of the rectangle is represented by 5
x
2
+ 19
x
+ 12.
What is the length?
5
x
+ 4
30.
Multiply.
31.
Multiply.
32.
Divide.
33.
Divide.
34.
Perform the indicated operations.
35.
Find the area of the rectangle shown.
36.
Subtract.
Express your answer in simplest form.
37.
Subtract.
Express your answer in simplest form.
38.
Add.
Express your answer in simplest form.
39.
Add.
Express yo.
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4. Important Idea
Many of the methods and
ideas we learned about
algebraic equations also
work with trigonometric
equations.
5. Example
1
cos
2
x =
Solve by
graphing:
1 cosy x=2. Graph
3. Graph 2
1
2
y =
4. Find window
5. Use 2nd
Calc 5 to
find 2 solutions
6. Use Period to
find all solutions
1. Set mode to rads
using
intersect
ion
method
6. Important Ideaand
always have 2
positive or 2
negative values
between 0 and
sin x cosx
2π . Find the 1st
2 solutions after
0 with your
calculator.
2π
8. Important Idea
will have
only 1 solution
between 0 and
tan x
(its period).
Find the 1st
solution after 0;
all other
solutions are
repeats.
π
0 π
9. Example
Solve
sin .85x = −
1. Set mode to rads
2. Re-write
3. Graph
1 sin .85y x= +
4. Find window
5. Find 2 solutions
using 2nd
calc 2
6. Use period to
find all solutions
using the
x-
intercept
method
11. Example
Solve
2
3sin cos 2 0x x− − =
using the x
intercept method.
Hint: find all
solutions within
the period and add
2 kπ to each.
12. Important Idea
1. Write in form ( ) 0f x =
2. Determine period, p
3. Graph over interval p
4. Graphically, find all
solutions in interval p
5. Add to solutionkp
To solve any trig equation:
