This document provides an overview of trigonometry including definitions of basic trigonometric functions, identities, graphs of trig functions, inverse trig functions, laws of sines and cosines, and vectors. It defines angles, conversions between degrees and radians, trig functions of right triangles, trig identities, graphs of sine, cosine, tangent and cotangent waves, inverse trig functions, and vector operations like addition, subtraction, scalar multiplication and dot products. The document is a tutorial covering the essential topics of trigonometry.
with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.
with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.
Digital Nativity: Education in the Generation of the Tech-SaavyChris Mogensen
"The newest generation of learners arriving at our shores have never been without technology in their lives…how does this simple fact change their perception of education? What does it mean for them, and us? Explore the paradigm of teaching to the Digital Native."
Presentation given at the Association of Adult Educators conference on October 23rd, 2015 at Nova Scotia Community College - Waterfront Campus in Dartmouth, Nova Scotia, Canada.
Bibliography available on request.
Digital Nativity: Education in the Generation of the Tech-SaavyChris Mogensen
"The newest generation of learners arriving at our shores have never been without technology in their lives…how does this simple fact change their perception of education? What does it mean for them, and us? Explore the paradigm of teaching to the Digital Native."
Presentation given at the Association of Adult Educators conference on October 23rd, 2015 at Nova Scotia Community College - Waterfront Campus in Dartmouth, Nova Scotia, Canada.
Bibliography available on request.
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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
Using blurred images to assess damage in bridge structures?Alessandro Palmeri
Faster trains and augmented traffic have significantly increased the number and amplitude of loading cycles experienced on a daily basis by composite steel-concrete bridges. This higher demand accelerates the occurrence of damage in the shear connectors between the two materials, which in turn can severely affect performance and reliability of these structures. The aim of this talk is to present the preliminary results of theoretical and experimental investigations undertaken to assess the feasibility of using the envelope of deflections and rotations induced by moving loads as a practical and cost-effective alternative to traditional methods of health monitoring for composite bridges. Both analytical and numerical formulations for this dynamic problem are presented and the results of a parametric study are discussed. A novel photogrammetric approach is also introduced, which allows identifying vibration patterns in civil engineering structures by analysing blurred targets in long-exposure digital images. The initial experimental validation of this approach is presented and further challenges are highlighted.
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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.
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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.
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Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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2. Angles, Arc length, Conversions
Angle measured in standard position.Angle measured in standard position.
Initial side is the positive x – axis which is fixed.Initial side is the positive x – axis which is fixed.
Terminal side is the ray in quadrant II, which is freeTerminal side is the ray in quadrant II, which is free
to rotate about the origin. Counterclockwise rotationto rotate about the origin. Counterclockwise rotation
is positive, clockwise rotation is negative.is positive, clockwise rotation is negative.
Coterminal Angles: Angles that have the same terminal side.Coterminal Angles: Angles that have the same terminal side.
60°, 420°, and –300° are all coterminal.60°, 420°, and –300° are all coterminal.
Degrees to radians: Multiply angle byDegrees to radians: Multiply angle by .
180
π
3180
60
ππ
=×
radiansradians
Radians to degrees: Multiply angle byRadians to degrees: Multiply angle by .
180
π
45
180
4
=×
π
π
Arc length = central angle x radius, orArc length = central angle x radius, or .rs θ=
Note: The central angle must be in radian measure.Note: The central angle must be in radian measure.
Note: 1 revolution = 360° = 2π radians.Note: 1 revolution = 360° = 2π radians.
3. Right Triangle Trig Definitions
• sin(A) = sine of A = opposite / hypotenuse = a/c
• cos(A) = cosine of A = adjacent / hypotenuse = b/c
• tan(A) = tangent of A = opposite / adjacent = a/b
• csc(A) = cosecant of A = hypotenuse / opposite = c/a
• sec(A) = secant of A = hypotenuse / adjacent = c/b
• cot(A) = cotangent of A = adjacent / opposite = b/a
A
a
b
c
B
C
5. Basic Trigonometric Identities
)cos(
)sin(
)tan(
A
A
A =
)sin(
)cos(
)cot(
A
A
A =
)csc(
1
)sin(
)sin(
1
)csc(
A
A
A
A
=
=
)sec(
1
)cos(
)cos(
1
)sec(
A
A
A
A
=
=
)cot(
1
)tan(
)tan(
1
)cot(
A
A
A
A
=
=
1)(cos)(sin 22
=+ AA
)(sec1)(tan 22
AA =+ )(csc)(cot1 22
AA =+
Quotient identities:
Reciprocal Identities:
Pythagorean Identities:
Even/Odd identities:
)csc()csc(
)sin()sin(
AA
AA
−=−
−=−
)cot()cot(
)tan()tan(
AA
AA
−=−
−=−
)sec()sec(
)cos()cos(
AA
AA
=−
=−
Even functions Odd functions Odd functions
6. AAll SStudents TTake CCalculus.
Quad II
Quad I
Quad III Quad IV
cos(A)>0
sin(A)>0
tan(A)>0
sec(A)>0
csc(A)>0
cot(A)>0
cos(A)<0
sin(A)>0
tan(A)<0
sec(A)<0
csc(A)>0
cot(A)<0
cos(A)<0
sin(A)<0
tan(A)>0
sec(A)<0
csc(A)<0
cot(A)>0
cos(A)>0
sin(A)<0
tan(A)<0
sec(A)>0
csc(A)<0
cot(A)<0
8. Unit circle
• Radius of the circle is 1.
• x = cos(θ)
• y = sin(θ)
• Pythagorean Theorem:
• This gives the identity:
• Zeros of sin(θ) are where n is an integer.
• Zeros of cos(θ) are where n is an
integer.
1)sin(1 ≤≤− θ
1)cos(1 ≤≤− θ
122
=+ yx
1)(sin)(cos 22
=+ θθ
πn
π
π
n+
2
9.
10. Graphs of sine & cosine
• Fundamental period of sine and cosine is 2π.
• Domain of sine and cosine is
• Range of sine and cosine is [–|A|+D, |A|+D].
• The amplitude of a sine and cosine graph is |A|.
• The vertical shift or average value of sine and
cosine graph is D.
• The period of sine and cosine graph is
• The phase shift or horizontal shift is
DCBxAxg
DCBxAxf
+−=
+−=
)cos()(
)sin()(
.
2
B
π
.
B
C
.ℜ
11. Sine graphs
y = sin(x)
y = sin(3x)
y = 3sin(x)
y = sin(x – 3)
y = sin(x) + 3
y = 3sin(3x-9)+3
y = sin(x)
y = sin(x/3)
12. Graphs of cosine
y = cos(x)
y = cos(3x)
y = cos(x – 3)
y = 3cos(x)
y = cos(x) + 3
y = 3cos(3x – 9) + 3
y = cos(x)
y = cos(x/3)
13. Tangent and cotangent graphs
• Fundamental period of tangent and cotangent is
π.
• Domain of tangent is where n is an
integer.
• Domain of cotangent where n is an
integer.
• Range of tangent and cotangent is
• The period of tangent or cotangent graph is
DCBxAxg
DCBxAxf
+−=
+−=
)cot()(
)tan()(
+≠ π
π
nxx
2
|
{ }πnxx ≠|
.ℜ
.
B
π
14. Graphs of tangent and cotangent
y = tan(x)
Vertical asymptotes at
y = cot(x)
Verrical asymptotes at .πnx =.
2
π
π
nx +=
15. Graphs of secant and cosecant
y = sec(x)
Vertical asymptotes at
Range: (–∞, –1] U [1, ∞)
y = cos(x)
y = csc(x)
Vertical asymptotes at
Range: (–∞, –1] U [1, ∞)
y = sin(x)
.
2
π
π
nx += .πnx =
16. Inverse Trigonometric Functions
and Trig Equations
)arctan()(tan 1
xxy == −
)arcsin()(sin 1
xxy == −
)arccos()(cos 1
xxy == −
−
2
,
2
ππ
Domain: [–1, 1]
Range:
0 < y < 1, solutions in QI and QII.
–1 < y < 0, solutions in QIII and QIV.
Domain: [–1, 1]
Range: [0, π]
0 < y < 1, solutions in QI and QIV.
–1< y < 0, solutions in QII and QIII.
−
2
,
2
ππ
Domain:
Range:
0 < y < 1, solutions in QI and QIII.
–1 < y < 0, solutions in QII and QIV.
ℜ
17. Trigonometric Identities
Summation & Difference Formulas
)tan()tan(1
)tan()tan(
)tan(
)sin()sin()cos()cos()cos(
)sin()cos()cos()sin()sin(
BA
BA
BA
BABABA
BABABA
±
=±
=±
±=±
18. Trigonometric Identities
Double Angle Formulas
)(tan1
)tan(2
)2tan(
1)(cos2)(sin21)(sin)(cos)2cos(
)cos()sin(2)2sin(
2
2222
A
A
A
AAAAA
AAA
−
=
−=−=−=
=
19. Trigonometric Identities
Half Angle Formulas
)cos(1
)cos(1
2
tan
2
)cos(1
2
cos
2
)cos(1
2
sin
A
AA
AA
AA
+
−
±=
+
±=
−
±=
The quadrant of 2
A
determines the sign.
20. Law of Sines & Law of Cosines
)sin()sin()sin(
)sin()sin()sin(
C
c
B
b
A
a
c
C
b
B
a
A
==
==
)cos(2
)cos(2
)cos(2
222
222
222
Abccba
Baccab
Cabbac
−+=
−+=
−+=
Law of sines Law of cosines
Use when you have a
complete ratio: SSA.
Use when you have SAS, SSS.
21. Vectors
• A vector is an object that has a magnitude and a direction.
• Given two points P1: and P2: on the plane, a
vector v that connects the points from P1 to P2 is
v = i + j.
• Unit vectors are vectors of length 1.
• i is the unit vector in the x direction.
• j is the unit vector in the y direction.
• A unit vector in the direction of v is v/||v||
• A vector v can be represented in component form
by v = vxi + vyj.
• The magnitude of v is ||v|| =
• Using the angle that the vector makes with x-axis in
standard position and the vector’s magnitude, component
form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j
22
yx vv +
),( 11 yx ),( 22 yx
)( 12 xx − )( 12 yy −
22. Vector Operations
Scalar multiplication: A vector can be multiplied by any scalar (or number).
Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j.
Dot Product: Multiplication of two
vectors.
Let v = vxi + vyj, w = wxi + wyj.
v · w = vxwx + vywy
Example: Let v = 5i + 4j, w = –2i + 3j.
v · w = (5)(–2) + (4)(3) = –10 + 12 = 2.
Two vectors v and w are orthogonal (perpendicular) iff v · w = 0.
Addition/subtraction of vectors: Add/subtract same components.
Example Let v = 5i + 4j, w = –2i + 3j.
v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j.
3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j.
||3v – 2w|| = 9.19397619 22
≈=+
Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the
angle between the two vectors.
θ
w
v