The document discusses trigonometry and circular functions. It defines trigonometry as the study of repeating, cyclical patterns through the use of trigonometric functions. It then provides information about measuring angles in degrees and radians, defining standard angle position, discussing coterminal angles, and converting between degree and radian measure.
CIRCLE
DEFINITION AND PROPERTIES OF A CIRCLE
A circle can be defined in two ways.
A circle: Is a closed path curve all points of which are equal-distance from a fixed point called centre OR
- Is a locus at a point which moves in a plane so that it is always of constant distance from a fixed point known as a centre.
CIRCLE
DEFINITION AND PROPERTIES OF A CIRCLE
A circle can be defined in two ways.
A circle: Is a closed path curve all points of which are equal-distance from a fixed point called centre OR
- Is a locus at a point which moves in a plane so that it is always of constant distance from a fixed point known as a centre.
trigonometric system lesson of math on how to. solve triangle the unit cirlce is the guide to find the exact value of a triangle,it is the foundation on how to rely the exact value of pi ..finding the sin the cosine the tangent the secant the cosecant and the cotangent
trigonometric system lesson of math on how to. solve triangle the unit cirlce is the guide to find the exact value of a triangle,it is the foundation on how to rely the exact value of pi ..finding the sin the cosine the tangent the secant the cosecant and the cotangent
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
1. Botanical Name
Narcissus 'Trigonometry'
Plant Common Name
Trigonometry Daffodil
The flowers of a Trigonometry Daffodil are of almost geometric
precision with their repeating patterns.
Repeating patterns occur in sound, light, tides, time, and nature.
To analyse these repeating, cyclical patterns, we need to study the
cyclical functions branch of trigonometry.
Math 30-1 1
4. Circular Functions
Angles can be measured in:
Degrees: common unit used in Geometry
1
part of a circle
360
Radian: common unit used in Trigonometry
1
part of a circle
2
Gradient: not common unit, used in surveying
1
part of a circle
400
Revolutions: angular velocity radians per second
Math 30-1 4
5. Angles in Standard Position
Initial arm
Vertex
Terminal
arm
x
y
To study circular functions, we must consider angles of rotation.
Math 30-1 5
6. If the terminal arm
moves counter-
clockwise, angle A
is positive.
A x
y
If the terminal side
moves clockwise,
angle A is
negative.
A
x
y
Positive or Negative Rotation Angle
McGraw Hill DVD Teacher Resources 4.1_178_IA
Math 30-1 6
8. Sketch each rotation angle in standard position.
State the quadrant in which the terminal arm lies.
400° - 170°
-1020°
1280°
Math 30-1 8
9. Coterminal angles are angles in standard position that share the
same terminal arm. They also share the same reference angle.
McGraw Hill DVD Teacher Resources 4.1_178_IA
50°
Rotation Angle 50°
Terminal arm is in quadrant I
Positive Coterminal Angles
Counterclockwise
50° + (360°)(1) =
Negative Coterminal Angles
Clockwise
-310°
770°
-670°
410°
50° + (360°)(2) =
50° + (360°)(-1) =
50° + (360°)(-2) =
Math 30-1 9
10. Coterminal Angles in General Form
By adding or subtracting multiples of one full rotation, you
can write an infinite number of angles that are coterminal
with any given angle.
θ ± (360°)n, where n is any natural number
Why must n be a natural number?
Math 30-1 10
11. Sketching Angles and Listing Coterminal Angles
Sketch the following angles in standard position. Identify all coterminal angles
within the domain -720° < θ < 720° . Express each angle in general form.
a) 1500 b) -2400
c) 5700
Positive
Negative
General Form
5100
-2100
1200
-6000
2100
-1500
150 360 ,
n n N
240 360 ,
n n N
570 360 ,
n n N
Positive
Negative
General Form
Positive
Negative
General Form
, -5700
, 4800
-5100
Math 30-1 11
12. Radian Measure: Trig and Calculus
The radian measure of an angle is the ratio of arc length of a
sector to the radius of the circle.
a
r
number of radians =
arc length
radius
When arc length = radius, the
angle measures one radian.
How many radians do you
think there are in one circle?
Math 30-1 12
13. Construct arcs on the
circle that are equal in
length to the radius.
Radian Measure
2 6.283185307...
radians
C 2r
arc length 2(1)
http://www.geogebra.org/en/upload/files/ppsb/radian.html
One full revolution is
Math 30-1 13
14. Radian Measure
One radian is the measure of the central angle subtended in a
circle by an arc of equal length to the radius.
2r
r
r
=
a
r
O r
r s = r
1 radian
= 1 revolution of 360
Therefore, 2π rad = 3600.
Or, π rad = 1800.
r
2 rads
Angle
measures
without
units are
considered
to be in
radians.
Math 30-1 14
17. Sketching Angles and Listing Coterminal Angles
Sketch the following angles in standard position. Identify all coterminal angles
within the domain -4π < θ < 4π . Express each angle in general form.
a) b) c)
Positive
Negative
General Form
5
2 ,
6
n n N
4
2 ,
3
n n N
10.47 2 ,
n n N
Positive
Negative
General Form
Positive
Negative
General Form
5
6
4
3
10.47
17
6
7
6
19
,
6
2
3
8
,
3
10
3
4.19
2.1
, 8.38
Math 30-1 17