This document discusses using the Law of Sines and Law of Cosines to solve oblique triangles. It covers the four cases for solving triangles: two angles and a side (AAS/ASA), two sides and an angle opposite (SSA), three sides (SSS), and two sides and their included angle (SAS). The Law of Sines can be used for AAS/ASA and SSA cases, while the Law of Cosines is needed for SSS and SAS cases. It also discusses finding the area of triangles using the Law of Sines and Heron's formula for SSS cases.
This learner's module discusses about the Six Trigonometric Ratios. It also teaches about the definition and characteristics of each of the Six Trigonometric Ratio.
This learner's module discusses about the Six Trigonometric Ratios. It also teaches about the definition and characteristics of each of the Six Trigonometric Ratio.
Math Specialization - 20 items I LET ReviewerFlipped Channel
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This learner's module talks about the topic Reasoning. It also includes the definition of Reasoning, Types of Reasoning (Inductive and Deductive Reasoning) and Examples of Reasoning for each type of reasoning.
From: 21st Century Lessons: A Boston Teachers Union Initiative and Corey Cheever. Use this Common Core State Standards aligned lesson to engage middle school math students with learning about identifying the slope of a line, and graphing a line with a given slope. The "Do Now" will remind students about the order of operations when dealing with negative numbers and fraction bars. Then, the students will see a demonstration of positive, negative, zero, and undefined slope. During the exploration, students will find slope by definition (rise/run), and the practice will turn towards the slope formula. Finally, the homework assignment investigates slope with regards to geometry. Find this linear equation lesson and companion worksheets - all free - on Share My Lesson: http://www.sharemylesson.com/teaching-resource/the-slope-of-a-line-50033011/
Math Specialization - 20 items I LET ReviewerFlipped Channel
If you happen to like this powerpoint, you may contact me at flippedchannel@gmail.com
I offer some educational services like:
-powerpoint presentation maker
-grammarian
-content creator
-layout designer
Subscribe to our platforms:
FlippED Channel (Youtube)
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LET in the NET (Facebook)
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This learner's module talks about the topic Reasoning. It also includes the definition of Reasoning, Types of Reasoning (Inductive and Deductive Reasoning) and Examples of Reasoning for each type of reasoning.
From: 21st Century Lessons: A Boston Teachers Union Initiative and Corey Cheever. Use this Common Core State Standards aligned lesson to engage middle school math students with learning about identifying the slope of a line, and graphing a line with a given slope. The "Do Now" will remind students about the order of operations when dealing with negative numbers and fraction bars. Then, the students will see a demonstration of positive, negative, zero, and undefined slope. During the exploration, students will find slope by definition (rise/run), and the practice will turn towards the slope formula. Finally, the homework assignment investigates slope with regards to geometry. Find this linear equation lesson and companion worksheets - all free - on Share My Lesson: http://www.sharemylesson.com/teaching-resource/the-slope-of-a-line-50033011/
Law of Cosines.ppt Law of Cosines.ppt Law of Cosines.pptJakeMamala
The Law of Cosines is a trigonometric formula used in geometry to find the measure of an angle or the length of a side in a triangle. It's particularly useful for solving triangles that are not right-angled. The law is an extension of the Pythagorean theorem.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
2. Use the Law of Sines to solve oblique triangles
(AAS or ASA).
Use the Law of Sines to solve oblique triangles
(SSA).
Find the areas of oblique triangles.
Use the Law of Sines to model and solve real-
life problems.
What You Should Learn
3. Plan for the day
When to use law of sines and law of cosines
Applying the law of sines
Law of sines - Ambiguous Case
4. Introduction
In this section, we will solve
oblique triangles – triangles
that have no right angles.
As standard notation, the
angles of a triangle are labeled A, B, and C, and
their opposite sides are labeled a, b, and c.
To solve an oblique triangle, we need to know the
measure of at least one side and any two other
measures of the triangle—either two sides, two
angles, or one angle and one side.
5. Introduction
This breaks down into the following four cases:
1. Two angles and any side (AAS or ASA)
2. Two sides and an angle opposite one of them (SSA)
3. Three sides (SSS)
4. Two sides and their included angle (SAS)
The first two cases can be solved using the
Law of Sines, whereas the last two cases require the
Law of Cosines.
7. Given Two Angles and One Side – AAS
For the triangle below C = 102, B = 29, and
b = 28 feet. Find the remaining angle and
sides.
8. Example AAS - Solution
The third angle of the triangle is
A = 180 – B – C
= 180 – 29 – 102
= 49.
By the Law of Sines, you have
.
9. Example AAS – Solution
Using b = 28 produces
and
cont’d
10. Law of Sines
For non right triangles
Law of sines
Try this:
A
B
C
c
a
b
C
c
B
b
A
a
sin
sin
sin
mm
c
B
A o
o
45
,
67
,
43
11. Let’s look at this: Example 1
Given a triangle, demonstrate using the Law of
Sines that it is a valid triangle (numbers are
rounded to the nearest tenth so they may be
up to a tenth off):
a = 5 A = 40o
b = 7 B = 64.1o
c = 7.5 C = 75.9o
Is it valid??
12. And this: Example 2
Given a triangle, demonstrate using the Law of
Sines that it is a valid triangle (numbers are
rounded to the nearest tenth so they may be
up to a tenth off):
a = 5 A = 40o
b = 7 B = 115.9o
c = 3.2 C = 24.1o
Is it valid??
Why does this work?
13. Looking at these two examples
a = 5 A = 40o
b = 7 B = 64.1o
c = 7.5 C = 75.9o
a = 5 A = 40o
b = 7 B = 115.9o
c = 3.2 C = 24.1o
In both cases a, b and A are the same (two sides and
an angle) but they produced two different triangles
Why??
14. Here is what happened
a = 5 A = 40o
b = 7 B =
c = C =
What is sin 64.1?
What is sin 115.9?
What is the relationship between these two angles?
Remember the sine of an angle in the first quadrant
(acute: 0o – 90o) and second quadrant,
(obtuse: 90o – 180o)are the same!
)
8999
(.
sin
5
40
sin
7
sin
sin
7
40
sin
5
1
o
o
B
B
16. The Ambiguous Case (SSA)
In our first example we saw that two angles and one
side determine a unique triangle.
However, if two sides and one opposite angle
are given, three possible situations can
occur:
(1) no such triangle exists,
(2) one such triangle exists, or
(3) two distinct triangles may satisfy the conditions.
17. Back to these examples
Given two sides and an angle across
a = 5 A = 40o
b = 7 B = 64.1o
c = 7.5 C = 75.9o
a = 5 A = 40o
b = 7 B = 115.9o
c = 3.2 C = 24.1o
The Ambiguous Case
19. Ambiguous Case
1. Is it Law of Sines or Law of Cosines
1. Law of Cosines – solve based upon one solution
2. Law of Sines – go to #2
2. Law of Sines - Is it the SSA case? (Two sides and angle opposite)
1. No – not ambiguous, solve based upon one solution
2. Yes – go to #3.
3. Is the side opposite the angle the shortest side?
1. No – not ambiguous, solve based upon one solution
2. Yes – go to #4
4. Is the angle obtuse?
1. No – go to #5
2. Yes – no solution
5. Calculate the height of the triangle
height = the side not opposite the angle x the sine of the angle
1. If the side opposite the angle is shorter than the height – no solution
2. If the side opposite the angle is equal to the height – one solution
3. If the side opposite the angle is longer than the height – two
solutions
20. How many solutions are there?
1. A = 30o a = 5 b = 3
2. B = 50o a = 6 b = 5
3. C = 80o b = 5 c = 6
4. A = 40o a = 4 b = 8
5. a = 5 b = 4 c = 6
6. B = 20o a = 10 c = 15
7. A = 25o a = 3 b = 6
8. C = 75o a = 5 b = 3
1
2
0
2
1
1
1
1
21. Example – Single-Solution Case—SSA
For the triangle below, a = 22 inches, b = 12
inches, and A = 42. Find the remaining side
and angles.
One solution: a b
22. Example – Solution SSA
By the Law of Sines, you have
Reciprocal form
Multiply each side by b.
Substitute for A, a, and b.
B is acute.
)
( 22
42
sin
12
sin
B
)
(sin
sin
a
A
b
B
a
A
b
B sin
sin
o
B 41
.
21
23. Example – Solution SSA
Now, you can determine that
C 180 – 42 – 21.41
= 116.59.
Then, the remaining side is
cont’d
)
59
.
116
sin(
)
42
sin(
22
c
C
A
a
c sin
sin
A
a
C
c
sin
sin
inches
40
.
29
25. How many solutions are there?
1. A = 30o a = 5 b = 3
2. B = 50o a = 6 b = 5
3. C = 80o b = 5 c = 6
4. A = 40o a = 4 b = 8
5. a = 5 b = 4 c = 6
6. B = 20o a = 10 c = 15
7. A = 25o a = 3 b = 6
8. C = 75o a = 5 b = 3
1
2
1
0
1
1
2
1
Solve #2, 7
27. Area of an Oblique Triangle
The procedure used to prove the Law of Sines leads to
a simple formula for the area of an oblique triangle.
Referring to the triangles below, that each triangle
has a height of h = b sin A.
A is acute. A is obtuse.
28. Area of a Triangle - SAS
SAS – you know two sides: b, c and
the angle between: A
Remember area of a triangle is
½ base ● height
Base = b
Height = c ● sin A
Area = ½ bc(sinA)
A
B
C
c a
b
h
Looking at this from all three sides:
Area = ½ ab(sin C) = ½ ac(sin B) = ½ bc (sin A)
30. Example – Finding the Area of a Triangular Lot
Find the area of a triangular lot having two sides of
lengths 90 meters and 52 meters and an included
angle of 102.
Solution:
Consider a = 90 meters, b = 52 meters, and the
included angle C = 102
Then, the area of the triangle is
Area = ½ ab sin C
= ½ (90)(52)(sin102)
2289 square meters.
33. Plan for the day
Law of Cosines
Finding the area of a triangle – Heron’s
Formula
34. Use the Law of Cosines to solve oblique
triangles (SSS or SAS).
Use the Law of Cosines to model and solve
real-life problems.
Use Heron’s Area Formula to find the area of a
triangle.
What You Should Learn
35. Introduction
Four cases.
1. Two angles and any side (AAS or ASA)
2. Two sides and an angle opposite one of them (SSA)
3. Three sides (SSS)
4. Two sides and their included angle (SAS)
The first two cases can be solved using the
Law of Sines, whereas the last two cases require the
Law of Cosines.
36. Law of Sines
For non right triangles
Law of sines
A
B
C
c
a
b
C
c
B
b
A
a
sin
sin
sin
37. Ambiguous Case
1. Is it Law or Sines or Law of Cosines
1. Law of Cosines – solve based upon one solution
2. Law of Sines – go to #2
2. Law of Sines - Is it the SSA case? (Two sides and angle opposite)
1. No – not ambiguous, solve based upon one solution
2. Yes – go to #3.
3. Is the side opposite the angle the shortest side?
1. No – not ambiguous, solve based upon one solution
2. Yes – go to #4
4. Is the angle obtuse?
1. No – go to #5
2. Yes – no solution
5. Calculate the height of the triangle
height = the side not opposite the angle x the sine of the angle
1. If the side opposite the angle is shorter than the height – no solution
2. If the side opposite the angle is equal to the height – one solution
3. If the side opposite the angle is longer than the height – two
solutions
38. Area of a Triangle - SAS
SAS – you know two sides: b, c and
the angle between: A
Remember area of a triangle is
½ base ● height
Base = b
Height = c ● sin A
Area = ½ bc(sinA)
A
B
C
c a
b
h
Looking at this from all three sides:
Area = ½ ab(sin C) = ½ ac(sin B) = ½ bc(sin A)
39. Law of Cosines: Introduction
Two cases remain in the list of conditions
needed to solve an oblique triangle – SSS
and SAS.
If you are given three sides (SSS), or two sides
and their included angle (SAS), none of the
ratios in the Law of Sines would be complete.
In such cases, you can use the Law of
Cosines.
40. Law of Cosines
Side, Angle, Side
A
B
C
c a
b
C
abCos
b
a
c
B
acCos
c
a
b
A
bcCos
c
b
a
2
2
2
2
2
2
2
2
2
2
2
2
41. Try these
1. B = 20o a = 10 c = 15
2. A = 25o b = 3 c = 6
3. C = 75o a = 5 b = 3
43. Law of Cosines
SSS
ab
c
b
a
C
ac
b
c
a
B
bc
a
c
b
A
2
cos
2
cos
2
cos
2
2
2
2
2
2
2
2
2
Always solve for the angle across from the longest side first!
44. Why
It is wise to find the largest angle when you
have SSS. Knowing the cosine of an angle,
you can determine whether the angle is acute
or obtuse. That is,
cos > 0 for 0 < < 90
cos < 0 for 90 < < 180.
This avoids the ambiguous case!
Acute
Obtuse
45. Try these
1. a = 5 b = 4 c = 6
2. a = 20 b = 10 c = 28
3. a = 8 b = 5 c = 12
47. An Application of the Law of Cosines
The pitcher’s mound on a
women’s softball field is 43 feet
from home plate and the
distance between the bases is
60 feet (The pitcher’s mound is
not halfway between home
plate and second base.) How
far is the pitcher’s mound from
first base?
48. Solution
In triangle HPF, H = 45 (line HP bisects the
right angle at H), f = 43, and p = 60.
Using the Law of Cosines for this SAS case,
you have
h2 = f2 + p2 – 2fp cos H
= 432 + 602 – 2(43)(60) cos 45
1800.3.
So, the approximate distance from the pitcher’s
mound to first base is
42.43 feet.
50. Heron’s Area Formula
The Law of Cosines can be used to establish
the following formula for the area of a
triangle. This formula is called Heron’s Area
Formula after the Greek mathematician
Heron (c. 100 B.C.).
51. Area of a Triangle
Law of Cosines Case - SSS
)
)(
)(
( c
s
b
s
a
s
s
A
A
B
C
c a
b
h
SSS – Given all three sides
Heron’s formula:
2
c
b
a
s
where
52. Try these
Given the triangle with three sides of 6, 8, 10
find the area
Given the triangle with three sides of 12, 15,
21 find the area