The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
This document provides an introduction to trigonometric ratios and their uses in solving problems involving right triangles. It defines the trig functions of sine, cosine, and tangent and relates them to angle measurements and the sides of a right triangle. Examples are given for calculating trig ratios and using them to find missing side lengths or angle measures. The document also distinguishes between angles of elevation and depression and provides examples of applying trigonometry to problems involving these angle types.
Students will learn to classify angles as right, acute, obtuse, or straight based on their measure. They will also learn about vertical, complementary and supplementary angles. Specifically, vertical angles are opposite angles formed by two intersecting lines that have the same measure, complementary angles add up to 90 degrees, and supplementary angles add up to 180 degrees. Students will practice using these concepts to find missing angle measures.
The document discusses trigonometric ratios in right triangles. It states that if two angles of one right triangle are congruent to two angles of another right triangle, the triangles are similar and their corresponding ratios are equal. It defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) in terms of the sides of a right triangle, using both symbols and words. It also introduces the mnemonic SOHCAHTOA to remember the definitions.
This document discusses the tangent ratio and how to use it to solve problems involving right triangles. It provides examples of finding the tangent of various angles, using the tangent ratio to find missing side lengths of triangles, and calculating the angle of a road given its grade. Key terms introduced are tangent ratio, sine, cosine, acute angle, opposite leg, and adjacent leg. Examples are provided to illustrate using the tangent ratio to solve problems.
Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
This document discusses trigonometric functions and how to determine their values in right triangles. It defines the six trig functions as ratios of the sides of a right triangle with a point P on the hypotenuse. It provides examples of calculating the trig functions when given a point P on the hypotenuse. It also discusses how to determine the quadrant of an angle based on the signs of various trig functions.
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
This document provides an introduction to trigonometric ratios and their uses in solving problems involving right triangles. It defines the trig functions of sine, cosine, and tangent and relates them to angle measurements and the sides of a right triangle. Examples are given for calculating trig ratios and using them to find missing side lengths or angle measures. The document also distinguishes between angles of elevation and depression and provides examples of applying trigonometry to problems involving these angle types.
Students will learn to classify angles as right, acute, obtuse, or straight based on their measure. They will also learn about vertical, complementary and supplementary angles. Specifically, vertical angles are opposite angles formed by two intersecting lines that have the same measure, complementary angles add up to 90 degrees, and supplementary angles add up to 180 degrees. Students will practice using these concepts to find missing angle measures.
The document discusses trigonometric ratios in right triangles. It states that if two angles of one right triangle are congruent to two angles of another right triangle, the triangles are similar and their corresponding ratios are equal. It defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) in terms of the sides of a right triangle, using both symbols and words. It also introduces the mnemonic SOHCAHTOA to remember the definitions.
This document discusses the tangent ratio and how to use it to solve problems involving right triangles. It provides examples of finding the tangent of various angles, using the tangent ratio to find missing side lengths of triangles, and calculating the angle of a road given its grade. Key terms introduced are tangent ratio, sine, cosine, acute angle, opposite leg, and adjacent leg. Examples are provided to illustrate using the tangent ratio to solve problems.
Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
This document discusses trigonometric functions and how to determine their values in right triangles. It defines the six trig functions as ratios of the sides of a right triangle with a point P on the hypotenuse. It provides examples of calculating the trig functions when given a point P on the hypotenuse. It also discusses how to determine the quadrant of an angle based on the signs of various trig functions.
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
This document provides examples of using trigonometric ratios (sine, cosine, tangent) to solve for unknown sides of right triangles when given angle and side measurements. Several examples are worked through, identifying opposite, adjacent, and hypotenuse sides based on a given angle and using trig functions (sine, cosine, tangent) to calculate values or unknown sides. The document demonstrates setting up and solving trigonometric ratios to find missing values in right triangles.
Trigonometric functions in standard position slide 1Jessica Garcia
This document discusses trigonometric functions of angles in standard position. It defines standard position as having the angle's vertex at the origin with one ray on the positive x-axis. Examples are provided for finding the reference angle of given angles and calculating trigonometric functions based on the coordinates of a point on the terminal side of the angle. Practice problems are included to draw angles in standard position and find trig values based on given angle measures or point coordinates.
This document introduces basic trigonometry concepts including:
- The parts of a right triangle - adjacent side, opposite side, and hypotenuse.
- How to identify the adjacent and opposite sides depending on the angle being considered.
- The three basic trigonometric ratios - sine, cosine, and tangent - defined in terms of the sides of a right triangle.
- The mnemonic device "SOHCAHTOA" for remembering the ratios.
- Examples showing how to calculate the trig ratios for angles in right triangles.
- How trig ratios are the same for similar triangles since corresponding angles are equal.
The document provides information on trigonometric ratios and how to use them to solve for missing angles and sides of right triangles. It defines the sine, cosine, and tangent ratios using opposite, adjacent, and hypotenuse side lengths relative to a designated angle θ. It gives examples of setting up and evaluating trig ratios, finding missing sides and angles of triangles using inverse trig functions, and solving application problems involving angles of elevation/depression.
1. The document discusses trigonometric ratios and how to use them to solve for missing side lengths and angle measures in right triangles.
2. It provides examples of setting up trig ratios, using the Pythagorean theorem, and using inverse trig functions to find missing angles.
3. The key steps are to label the sides of the right triangle, set up the appropriate trig ratios based on which information is known or missing, and use trig identities or the inverse functions to calculate the missing information.
This is a basic intoductory unit on trigonometry meant for high school students in geometry. It is aligned to the Common Core States Standards covering right triangular geometry.
The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY".
It includes the --
1) Definition of Trigonometry,
2) History of Trigonometry and its Etymology,
3) Angles of a Right Triangle,
4) About different Trigonometric Ratios,
5) Some useful Mnemonics to remember the Trig. ratios,
6) Theorem, which states that --
"Trigonometric Ratios are same for the same angles"
7) Trigonometric Ratios for some specific/ standard angles.
This document introduces the SOH CAH TOA mnemonic for remembering the definitions of the trigonometric ratios sine, cosine, and tangent. It states that SOH CAH TOA is a trigonometric ratio and defines sine as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. It then provides examples of using trigonometric ratios to solve for unknown angles and sides of right triangles.
This document is from a geometry textbook and covers trigonometric ratios. It begins with examples of finding sine, cosine, and tangent values for various angles through using special right triangles, calculators, and trigonometric ratio definitions. It then demonstrates using trigonometric ratios to calculate unknown side lengths in right triangles when given one side and an angle. Word problems are also presented where students must set up and solve equations to find distances. The document concludes with a lesson quiz reviewing the material.
The document discusses trigonometric ratios and their properties. It defines the three basic trigonometric ratios - sine, cosine, and tangent - for any angle. It also discusses reciprocal ratios like cosecant, secant, and cotangent, which are the reciprocals of the basic ratios. The document provides examples of calculating trigonometric ratios for various angles and triangles. It also examines ratios of complementary and special angles like 0, 30, 45, 60, 90 degrees.
The document discusses trigonometric ratios and right triangles. It defines trigonometric ratios like sine, cosine, and tangent using the sides of a right triangle. It also describes two special right triangles - the 30-60-90 triangle and the 45-45-90 triangle - that are used often in trigonometry.
Trigonometry is the study of triangles and their relationships. The document discusses how trigonometry is used in fields like architecture, astronomy, geology, and for measuring distances and heights. It provides examples of how trigonometry can be used to calculate the height of a building given the distance and angle of elevation to its top.
This document discusses trigonometric ratios and identities. It defines trigonometric ratios as relationships between sides and angles of a right triangle. Specific ratios are defined for angles of 0, 30, 45, 60, and 90 degrees. Complementary angle identities are examined, showing ratios are equal for complementary angles (e.g. sin(90-A)=cos(A)). Trigonometric identities are derived from the Pythagorean theorem, including cos^2(A) + sin^2(A) = 1, sec^2(A) = 1 + tan^2(A), and cot^2(A) + 1 = cosec^2(A). Examples are provided to demonstrate using identities when
Maths ppt on some applications of trignometryHarsh Mahajan
This document discusses trigonometry and how it can be used to calculate heights and distances. It defines trigonometric ratios and the angles of elevation and depression. It then provides examples of using trigonometry to calculate the height of a tower given the angle of elevation is 30 degrees and the distance from the observer is 30 meters. It also gives an example of calculating the height of a pole using the angle made by the rope and the ground.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
This document provides examples of using trigonometric ratios (sine, cosine, tangent) to solve for unknown sides of right triangles when given angle and side measurements. Several examples are worked through, identifying opposite, adjacent, and hypotenuse sides based on a given angle and using trig functions (sine, cosine, tangent) to calculate values or unknown sides. The document demonstrates setting up and solving trigonometric ratios to find missing values in right triangles.
Trigonometric functions in standard position slide 1Jessica Garcia
This document discusses trigonometric functions of angles in standard position. It defines standard position as having the angle's vertex at the origin with one ray on the positive x-axis. Examples are provided for finding the reference angle of given angles and calculating trigonometric functions based on the coordinates of a point on the terminal side of the angle. Practice problems are included to draw angles in standard position and find trig values based on given angle measures or point coordinates.
This document introduces basic trigonometry concepts including:
- The parts of a right triangle - adjacent side, opposite side, and hypotenuse.
- How to identify the adjacent and opposite sides depending on the angle being considered.
- The three basic trigonometric ratios - sine, cosine, and tangent - defined in terms of the sides of a right triangle.
- The mnemonic device "SOHCAHTOA" for remembering the ratios.
- Examples showing how to calculate the trig ratios for angles in right triangles.
- How trig ratios are the same for similar triangles since corresponding angles are equal.
The document provides information on trigonometric ratios and how to use them to solve for missing angles and sides of right triangles. It defines the sine, cosine, and tangent ratios using opposite, adjacent, and hypotenuse side lengths relative to a designated angle θ. It gives examples of setting up and evaluating trig ratios, finding missing sides and angles of triangles using inverse trig functions, and solving application problems involving angles of elevation/depression.
1. The document discusses trigonometric ratios and how to use them to solve for missing side lengths and angle measures in right triangles.
2. It provides examples of setting up trig ratios, using the Pythagorean theorem, and using inverse trig functions to find missing angles.
3. The key steps are to label the sides of the right triangle, set up the appropriate trig ratios based on which information is known or missing, and use trig identities or the inverse functions to calculate the missing information.
This is a basic intoductory unit on trigonometry meant for high school students in geometry. It is aligned to the Common Core States Standards covering right triangular geometry.
The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY".
It includes the --
1) Definition of Trigonometry,
2) History of Trigonometry and its Etymology,
3) Angles of a Right Triangle,
4) About different Trigonometric Ratios,
5) Some useful Mnemonics to remember the Trig. ratios,
6) Theorem, which states that --
"Trigonometric Ratios are same for the same angles"
7) Trigonometric Ratios for some specific/ standard angles.
This document introduces the SOH CAH TOA mnemonic for remembering the definitions of the trigonometric ratios sine, cosine, and tangent. It states that SOH CAH TOA is a trigonometric ratio and defines sine as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. It then provides examples of using trigonometric ratios to solve for unknown angles and sides of right triangles.
This document is from a geometry textbook and covers trigonometric ratios. It begins with examples of finding sine, cosine, and tangent values for various angles through using special right triangles, calculators, and trigonometric ratio definitions. It then demonstrates using trigonometric ratios to calculate unknown side lengths in right triangles when given one side and an angle. Word problems are also presented where students must set up and solve equations to find distances. The document concludes with a lesson quiz reviewing the material.
The document discusses trigonometric ratios and their properties. It defines the three basic trigonometric ratios - sine, cosine, and tangent - for any angle. It also discusses reciprocal ratios like cosecant, secant, and cotangent, which are the reciprocals of the basic ratios. The document provides examples of calculating trigonometric ratios for various angles and triangles. It also examines ratios of complementary and special angles like 0, 30, 45, 60, 90 degrees.
The document discusses trigonometric ratios and right triangles. It defines trigonometric ratios like sine, cosine, and tangent using the sides of a right triangle. It also describes two special right triangles - the 30-60-90 triangle and the 45-45-90 triangle - that are used often in trigonometry.
Trigonometry is the study of triangles and their relationships. The document discusses how trigonometry is used in fields like architecture, astronomy, geology, and for measuring distances and heights. It provides examples of how trigonometry can be used to calculate the height of a building given the distance and angle of elevation to its top.
This document discusses trigonometric ratios and identities. It defines trigonometric ratios as relationships between sides and angles of a right triangle. Specific ratios are defined for angles of 0, 30, 45, 60, and 90 degrees. Complementary angle identities are examined, showing ratios are equal for complementary angles (e.g. sin(90-A)=cos(A)). Trigonometric identities are derived from the Pythagorean theorem, including cos^2(A) + sin^2(A) = 1, sec^2(A) = 1 + tan^2(A), and cot^2(A) + 1 = cosec^2(A). Examples are provided to demonstrate using identities when
Maths ppt on some applications of trignometryHarsh Mahajan
This document discusses trigonometry and how it can be used to calculate heights and distances. It defines trigonometric ratios and the angles of elevation and depression. It then provides examples of using trigonometry to calculate the height of a tower given the angle of elevation is 30 degrees and the distance from the observer is 30 meters. It also gives an example of calculating the height of a pole using the angle made by the rope and the ground.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
2. Mosque Detail, provided by Let’s Book (Jamie Carter)
on Flickr, Creative Commons, some rights reserved, April
2009, Seoul, Korea.
3. What are Trig Functions?
Let’s look at 3 similar triangles
If the angles of one triangle are exactly the
same as the angles of another, those
triangles are similar.
4. β 5
3
θ
4
β
7.5
4.5
θ
6
β
10
6
These are 3 similar right triangles;
they each have a 90° angle,
θ equal angles θ, and equal
8 angles β.
7. Ratios are the Same
Did you notice that each of the 3 similar
triangles, even though the triangles were
different sizes, had the exact same ratio
of sides?
15. Left Arrows Reverse Button, Provided by: Microsoft
How did you do?
Ifyou correctly identified the sine ratios in
those triangles, click to move to the next
section.
Otherwise, click below to go back.
Go to the beginning Review the sine ratio
16. Decorative and Calligraphic Tilewormk, provided by A.
Davey on flickr, Creative Commons, Some rights reserved,
April 2010, Mosque, Esfahan, Iran.
22. Left Arrows Reverse Button, Provided by: Microsoft
How did you do?
Ifyou correctly identified the cosine ratios
in those triangles, click to move to the
next section.
Otherwise, click below to go back.
Review the sine ratio Review the cosine ratio
23. 3D Ornamental Mosaic Tile, provided by A. Davey on flickr,
Creative Commons, Some rights reserved, April 2010, Mosque,
Kerman, Iran
29. Left Arrows Reverse Button, Provided by: Microsoft
How did you do?
Ifyou correctly identified the tangent
ratios in those triangles, click to move to
the next section.
Otherwise, click below to go back.
Review the cosine ratio Review the tangent ratio
30. SOH – CAH - TOA
You can remember the definitions of the 3
trig ratios with a mnemonic
SOH: Sin=Opposite/Hypotenuse
CAH: Cos=Adjacent/Hypotenuse
TOA: Tan=Opposite/Adjacent
31. Detail 3-D Tilework, provided by A. Davey on flickr, Creative
Commons, Some rights reserved, April 2010, Mosque,
Esfahan, Iran.
32. Detail 3-D Tilework, provided by A. Davey on flickr, Creative
Commons, Some rights reserved, April 2010, Mosque,
Esfahan, Iran.
Thanks for
Watching.
Any idea why I included
pictures from the Middle East?