The document discusses solving right triangles by using trigonometric ratios to find missing angles and sides given certain information like an angle measurement or side length. It also covers solving problems involving angles of elevation and depression by setting up trigonometric equations and solving for the unknown based on a sketch of the situation. Examples are provided to demonstrate these problem solving techniques step-by-step.
Chapter 1 ( Basic Concepts in Geometry )rey castro
Chapter 1 Basic Concepts in Geometry
1.1 Points, Lines and Planes
1.2 Line Segment
1.3 Rays and Angles
1.4 Some Special Angles
1.5 Angles Made By A Transversal
1.6 Transversal Across Two Parallel Lines
1.7 Conditions For Parallelism
This will help you in differentiating subsets of a line such as line segments, ray and opposite rays. Also in finding the number of line segments and rays in a given line.
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1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
Chapter 1 ( Basic Concepts in Geometry )rey castro
Chapter 1 Basic Concepts in Geometry
1.1 Points, Lines and Planes
1.2 Line Segment
1.3 Rays and Angles
1.4 Some Special Angles
1.5 Angles Made By A Transversal
1.6 Transversal Across Two Parallel Lines
1.7 Conditions For Parallelism
This will help you in differentiating subsets of a line such as line segments, ray and opposite rays. Also in finding the number of line segments and rays in a given line.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Embracing GenAI - A Strategic ImperativePeter 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.
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.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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.
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.
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
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2. Concepts and Objectives
⚫ Solve a right triangle given an angle and a side or two
sides.
⚫ Solve problems involving angles of elevation and
depression.
3. Solving Right Triangles
⚫ To solve a triangle means to find all of the missing
measures of the angles and sides.
⚫ There are two types of problems: right triangles with
⚫ a side and an angle or
⚫ two sides
⚫ We will use the trig ratios to find the missing pieces. The
key is to match the information we have and need to find
with a corresponding trig ratio.
4. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
5. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
The easiest part is to
calculate B by subtracting
A from 90°.
B = 90 – 34° 30′
= 55° 30′
6. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
To find the missing sides,
since we have the
hypotenuse, we will use
sine and cosine.
sin
a
A
c
=
7. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
To find the missing sides,
since we have the
hypotenuse, we will use
sine and cosine.
sin
a
A
c
=
sin34 30
12.7
a
=
8. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
To find the missing sides,
since we have the
hypotenuse, we will use
sine and cosine.
sin
a
A
c
=
sin34 30
12.7
a
= 12.7sin34 30
7.19 in
a =
=
9. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
To find the missing sides,
since we have the
hypotenuse, we will use
sine and cosine.
cos
b
A
c
=
cos34 30
12.7
b
=
10. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if A = 34° 30′ and
c = 12.7 in.
c = 12.7 in.
34° 30′
A
B
C
a
b
To find the missing sides,
since we have the
hypotenuse, we will use
sine and cosine.
cos
b
A
c
=
cos34 30
12.7
b
= 12.7cos34 30
10.5 in
b =
=
11. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if a = 29.43 cm and
c = 53.58 cm
c = 53.58 cm
a=29.43cm
A
B
C
b
This time, we’ll find the
missing side first, using the
Pythagorean Theorem.
2 2
b c a= −
2 2
53.58 29.43= −
44.77 cm=
12. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if a = 29.43 cm and
c = 53.58 cm
c = 53.58 cm
a=29.43cm
A
B
C
b
We can find A by using the
inverse of the sine function
because we have a and c.
sin
a
A
c
=
13. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if a = 29.43 cm and
c = 53.58 cm
c = 53.58 cm
a=29.43cm
A
B
C
b
We can find A by using the
inverse of the sine function
because we have a and c.
sin
a
A
c
=
1 29.43
sin
53.58
A −
=
14. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if a = 29.43 cm and
c = 53.58 cm
c = 53.58 cm
a=29.43cm
A
B
C
b
We can find A by using the
inverse of the sine function
because we have a and c.
sin
a
A
c
=
1 29.43
sin
53.58
A −
=
33.32A =
15. Solving Right Triangles (cont.)
⚫ Example: Solve right triangle ABC, if a = 29.43 cm and
c = 53.58 cm
c = 53.58 cm
a=29.43cm
A
B
C
b
We can find A by using the
inverse of the sine function
because we have a and c.
sin
a
A
c
=
1 29.43
sin
53.58
A −
=
33.32A =
90 33.32
56.68
B = −
=
16. Elevation and Depression
⚫ An angle of elevation is an angle formed by a horizontal
line and the line of sight to a point above the line.
⚫ An angle of depression is formed by a horizontal line and
a line of sight to a point below the line.
To identify whether an
angle is an angle of
elevation or depression,
check whether the line
of sight is above or
below the horizontal
line.
17. Elevation and Depression (cont.)
⚫ Example: If a tree casts a shadow 18 feet long when the
sun is at an elevation of 35˚, how tall is the tree to the
nearest foot?
18. Elevation and Depression (cont.)
⚫ Example: If a tree casts a shadow 18 feet long when the
sun is at an elevation of 35˚, how tall is the tree to the
nearest foot?
35˚
18´ (adj)
x
(opp)
Step 1: Draw a sketch
and label it.
19. Elevation and Depression (cont.)
⚫ Example: If a tree casts a shadow 18 feet long when the
sun is at an elevation of 35˚, how tall is the tree to the
nearest foot?
35˚
18´ (adj)
x
(opp)
tan35
18
x
=
Step 1: Draw a sketch
and label it.
Step 2: Use the sketch to
set up an equation.
20. Elevation and Depression (cont.)
⚫ Example: If a tree casts a shadow 18 feet long when the
sun is at an elevation of 35˚, how tall is the tree to the
nearest foot?
35˚
18´ (adj)
x
(opp)
tan35
18
x
=
Step 1: Draw a sketch
and label it.
18tan35
13 feet
x
x
=
=
Step 2: Use the sketch to
set up an equation.
Step 3: Solve the equation.
21. Elevation and Depression (cont.)
⚫ Example: A plane, at an altitude of 3000 feet, observes
the airport at an angle of 27°. What is the horizontal
distance between the plane and the airport to the
nearest foot?
22. Elevation and Depression (cont.)
⚫ Example: A plane, at an altitude of 3000 feet, observes
the airport at an angle of 27°. What is the horizontal
distance between the plane and the airport to the
nearest foot?
27°
3000′
x
Notice that the angle is
outside the triangle!
23. Elevation and Depression (cont.)
⚫ Example: A plane, at an altitude of 3000 feet, observes
the airport at an angle of 27°. What is the horizontal
distance between the plane and the airport to the
nearest foot?
27°
3000′
x
Notice that the angle is
outside the triangle!
Because horizontal lines
are parallel, we can use the
corresponding angle of
elevation.
(27°)
24. Elevation and Depression (cont.)
⚫ Example: A plane, at an altitude of 3000 feet, observes
the airport at an angle of 27°. What is the horizontal
distance between the plane and the airport to the
nearest foot?
27°
3000′
(opp)
x
(adj)
(27°)
3000
tan27
x
=
3000
5888 feet
tan27
x = =
